Classroom Tips and Techniques: Simultaneous Diagonalization and the Generalized Eigenvalue ProblemJSFHRobert J. LopezEmeritus Professor of Mathematics and Maple FellowMaplesoftJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=This article explores the connections between the generalized eigenvalue problem and the problem of simultaneously diagonalizing a pair of n \303\227 n matrices.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Given the n \303\227 n matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the generalized eigenvalue problem seeks the eigenpairs 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, solutions of the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GLDYnUSJ4RicvJSVib2xkR0YxL0YwUSZmYWxzZUYnL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0Y9LUkjbW9HRiQ2LVEiPUYnL0YzUSdub3JtYWxGJy8lJmZlbmNlR0Y7LyUqc2VwYXJhdG9yR0Y7LyUpc3RyZXRjaHlHRjsvJSpzeW1tZXRyaWNHRjsvJShsYXJnZW9wR0Y7LyUubW92YWJsZWxpbWl0c0dGOy8lJ2FjY2VudEdGOy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlYtRiw2JVEnJiM5NTU7RidGOkZELUZBNi1RIn5GJ0ZERkZGSEZKRkxGTkZQRlIvRlVRJjAuMGVtRicvRlhGam4tRiw2JVEiQkYnRi9GMkY1RkQ=, or 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. If LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is nonsingular, the eigenpairs of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYmLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkYyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiVRIkFGJ0YyRjVGPg== are solutions. If a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exists for which 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, and 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, where LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy is a diagonal matrix and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the n \303\227 n identity, then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are said to be diagonalized simultaneously, in which case the diagonal entries of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are the generalized eigenvalues for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=. Such a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= exists if LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is symmetric and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is positive definite. (Our definition of positive definite includes symmetry.)The Eigenvectors command in the LinearAlgebra package will return the generalized eigenpairs, which can also be obtained by a variety of brute-force calculations. There is single command in Maple for finding LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, the transition matrix for the simultaneous diagonalization problem. Note, finally, that in general, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is not an orthogonal matrix, so the transformations of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= to diagonal form are not similarity transformations.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Motivation</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Provided friction is ignored, the spring-mass system sketched in Figure 1 is modeled by the differential 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
PX := proc(h::{algebraic,procedure},g::{range,name=range})
local L, D, var;
if type(g,'range') then L := lhs(g); D := rhs(g) - L;
if not type(h,'procedure') then var := indets(h,'name');
if nops(var) <> 1 then error "need to specify a variable"; end if;
var := op(var); end if;
else L := lhs(rhs(g));D := rhs(rhs(g)) - L;var := lhs(g); end if;
if type(h,'procedure') then proc(x::algebraic) h(x - floor((x-L)/D)*D); end;
else proc(x::algebraic) eval(h, var = x - floor((x-L)/D)*D); end;
end if;
end:g := piecewise(x<1,x, x<3,2-x, x-4):
f := PX(g,x=0..4):
p1 := plot(f(x),x=0..12, color=black):
p2 := plot(f(x),x=16..28, color=black):
p3 := plot(f(x),x=32..44, color=black):
p4 := plot([[14,0]],style=point,symbol=solidcircle,symbolsize=45,color=black):
p5 := plot([[30,0]],style=point,symbol=solidcircle,symbolsize=45,color=black):
p6 := plot({[[12,0],[13,0]],[[15,0],[16,0]], [[28,0],[29,0]], [[31,0],[32,0]], [[44,0],[45,0]], [[-1,0],[0,0]]},x=-1..45,color=black):
p7 := plot({[[-1,-1.5],[-1,1.5]], [[45,-1.5],[45,1.5]]},color=black,thickness=2):
p8:=plots[textplot]({[14,2.3,typeset(m[1])],[30,2.3,typeset(m[2])],[6,2,typeset(k[1])],[22,2,typeset(k[2])],[38,2,typeset(k[3])],[16.5,-5,typeset(x[1])],[32.5,-5,typeset(x[2])]},font=[Times,Roman,12]):
p9 := plot({[[14,-2],[14,-4]], [[30,-2],[30,-4]]},color=black):
p10 := plottools[arrow]([14.5,-3],[20,-3],.1,.9,.3):
p11 := plottools[arrow]([30.5,-3],[36,-3],.1,.9,.3):
plots[display]([p||(1..11)],scaling=constrained,axes=none);Figure 1 Spring-mass system
These equations can be cast into the form 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, provided the following definitions are made.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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Simultaneous diagonalization of the matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiS0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= means that 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 and 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, so that 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 and 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. Hence, the differential equation 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 multiplying from the left by LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiZRIlRGJy8lJXNpemVHUSMxMEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnRkI=. Defining 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, or equivalently, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjctSSNtb0dGJDYtUSI9RicvRjZRJ25vcm1hbEYnLyUmZmVuY2VHRjQvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGUC1GLDYlUSJTRicvRjNGMS9GNlEnaXRhbGljRictRiw2J1EiWEYnRi9GMkY1RjhGPg==, puts the differential equation into the separated form 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, from which a solution is immediately available. The square roots of the diagonal elements in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy are the angular frequencies for the normal modes of oscillation of this undamped vibrating system.Alternatively, if solutions of the form 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 are sought, where v is a constant vector, the differential equation becomes 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, a generalized eigenvalue problem.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Example 1</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=If L0kiQUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIipHIypRWyI=assign( ); and L0kiQkc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIiozcmBpIg==, the eigenpairs for the generalized eigenvalue problem 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 and 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A matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that simultaneously diagonalizes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= 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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=That LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= are symmetric is obvious by inspection; that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is positive definite can be established by showing its eigenvalues are positive or by applying the IsDefinite command from LinearAlgebra.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Initialize Maple and enter the data:
Tools: Load Package\342\211\273Linear Algebrawith(LinearAlgebra):with(Student:-LinearAlgebra):Loading LinearAlgebrawith(LinearAlgebra):Control-drag the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSkmaGVsbGlwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZTRjk=Context Menu: Assign NameL0kiQUc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIipLOlJbIg==LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEnYXNzaWduRidGL0YyRkM=Control-drag the equation LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUScmbWxkcjtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTY2NjY2N2VtRicvRk5GU0Y5Context Menu: Assign NameL0kiQkc2Ii1JJ01hdHJpeEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJDYjL0kkJWlkR0YkIionZiJSWyI=LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEnYXNzaWduRidGL0YyRkM=assign( );
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Show that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is positive definite:
Type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and press the Enter key.Context Menu: Eigenvalues, etc\342\211\273EigenvaluesContext Menu: Approximate\342\211\2735SSJCRzYiprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEsZWlnZW52YWx1ZXNGJ0YvRjJGQw==LinearAlgebra:-Eigenvalues( );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEsYXR+NX5kaWdpdHNGJ0YvRjJGQw==evalf[5]( );Apply the IsDefinite command.LUkrSXNEZWZpbml0ZUc2IjYkSSJCR0YkL0kmcXVlcnlHRiRJMnBvc2l0aXZlX2RlZmluaXRlR0Yk = print(); # input placeholderJSFH
To obtain the eigenpairs that satisfy the generalized eigenvalue problem, use the Eigenvectors command:LUktRWlnZW52ZWN0b3JzRzYiNiRJIkFHRiRJIkJHRiQ= = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The vector on the right is a vector of eigenvalues. The columns in the matrix on the right are the corresponding eigenvectors. The order in which the eigenpairs are returned is not fixed. Repeated execution of the same command can reverse the order. The matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= given above is based on the order 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, 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.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">The Generalized Eigenvalue Problem</Text-field>JSFHThe Eigenvectors command has provided the generalized eigenpairs for the matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= in Example 1. These eigenpairs are also available as the eigenpairs of the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiRkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1GLDYlUSJCRidGL0YyLUYjNiYtRjY2LVEqJnVtaW51czA7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmVuLUkjbW5HRiQ2JFEiMUYnRjlGL0YyLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GLDYlUSJBRidGL0YyRjk=, as can be seen from LUktRWlnZW52ZWN0b3JzRzYiNiMtSTBkZWxheURvdFByb2R1Y3RHNiQlKnByb3RlY3RlZEcvSSttb2R1bGVuYW1lR0YkSSxUeXBlc2V0dGluZ0c2JEYpSShfc3lzbGliR0YkNiQqJEkiQkdGJCEiIkkiQUdGJA== = print(); # input placeholderJSFHAlternatively, the generalized eigenpairs can also be obtained from first principles with the following calculations.JSFH
Define the matrix 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Expression palette: 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print(); # input placeholderObtain and solve 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Type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRJyYjOTU1O0YnL0YwUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0Y/Rj9GPw== and press the Enter key.Context Menu: Standard Operations\342\211\273DeterminantContext Menu: Solve\342\211\273SolveLUkiQ0c2IjYjSSdsYW1iZGFHRiQ=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEsZGV0ZXJtaW5hbnRGJ0YvRjJGQw==LinearAlgebra:-Determinant( );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEmc29sdmVGJ0YvRjJGQw==solve( { = 0 } );
JSFHObtain the generalized eigenvector corresponding to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1GNjYtUSomdW1pbnVzMDtGJ0YyRjlGO0Y9Rj9GQUZDRkUvRkhRLDAuMjIyMjIyMmVtRicvRktGUC1JI21uR0YkNiRRIjFGJ0YyRjI=:
Type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZCLyUpc3RyZXRjaHlHRkIvJSpzeW1tZXRyaWNHRkIvJShsYXJnZW9wR0ZCLyUubW92YWJsZWxpbWl0c0dGQi8lJ2FjY2VudEdGQi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlEtSSNtbkdGJDYkUSIxRidGPkY+Rj5GPg== and press the Enter key.Context Menu: Vector Spaces\342\211\273Null SpaceLUkiQ0c2IjYjISIiprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVErbnVsbH5zcGFjZUYnRi9GMkZD[Student:-LinearAlgebra:-NullSpace( )];Eigenvector from first principlesEnter 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, where 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Context Menu: Conversions\342\211\273To ListContext Menu: Solve\342\211\273Solve for Variables\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Expression palette: Evaluation template
Evaluate x at the solution for aContext Menu: Evaluate at a Point\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkLUkiQ0dGKDYjISIiLUkkPCw+R0YoNiRJImFHRihJImJHRig=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEodG9+bGlzdEYnRi9GMkZDconvert( , 'list' );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc29sdmV+KHNwZWNpZmllZClGJ0YvRjJGQw==solve( , {a } );LUklZXZhbEclKnByb3RlY3RlZEc2JC1JJDwsPkdJKF9zeXNsaWJHNiI2JEkiYUdGKUkiYkdGKS9GKywkRiwjISIiIiIkprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyZXZhbHVhdGV+YXR+cG9pbnRGJ0YvRjJGQw==eval( , [b = 1]);LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
Obtain the generalized eigenvector corresponding to LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEnJiM5NTU7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRIzQzRidGMi1GNjYtUSIvRidGMkY5RjsvRj5RJXRydWVGJ0Y/RkFGQ0ZFL0ZIUSwwLjE2NjY2NjdlbUYnL0ZLRlYtRk02JFEjMTBGJ0YyRjI=:
Type LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEjNDNGJy9GM1Enbm9ybWFsRictSSNtb0dGJDYtUSIvRidGPi8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGRi8lKXN0cmV0Y2h5R0YxLyUqc3ltbWV0cmljR0ZGLyUobGFyZ2VvcEdGRi8lLm1vdmFibGVsaW1pdHNHRkYvJSdhY2NlbnRHRkYvJSdsc3BhY2VHUSwwLjE2NjY2NjdlbUYnLyUncnNwYWNlR0ZVLUY7NiRRIzEwRidGPkY+Rj5GPg== and press the Enter key.Context Menu: Vector Spaces\342\211\273Null SpaceLUkiQ0c2IjYjIyIjViIjNQ==print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVErbnVsbH5zcGFjZUYnRi9GMkZD[Student:-LinearAlgebra:-NullSpace( )];Eigenvector from first principlesEnter 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, where 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Context Menu: Conversions\342\211\273To ListContext Menu: Solve\342\211\273Solve for Variables\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Expression palette: Evaluation template
Evaluate x at the solution for aContext Menu: Evaluate at a Point\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiYkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkLUkiQ0dGKDYjIyIjViIjNS1JJDwsPkdGKzYkSSJhR0YoSSJiR0Yoprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEodG9+bGlzdEYnRi9GMkZDconvert( , 'list' );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyc29sdmV+KHNwZWNpZmllZClGJ0YvRjJGQw==solve( , {a } );LUklZXZhbEclKnByb3RlY3RlZEc2JC1JJDwsPkdJKF9zeXNsaWJHNiI2JEkiYUdGKUkiYkdGKS9GKywkRiwjISM+IiIlprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEyZXZhbHVhdGV+YXR+cG9pbnRGJ0YvRjJGQw==eval( , [b = 1]);
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Simultaneous Diagonalization</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The following prescription for finding a matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= that simultaneously diagonalizes LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is taken from [1, p.516], [2, p.1151], and [3, p.429].Let 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 be the eigenpairs of the matrix 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, where the LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EidkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2JS1GLzYlUSJrRicvRjZGNC9GOVEnaXRhbGljRidGQkZDLyUvc3Vic2NyaXB0c2hpZnRHUSIwRicvRjlRJ25vcm1hbEYn are unit vectors, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the lower triangular factor in the Cholesky decomposition 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; then LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYkLUZQNiUtRiw2JVEiTEYnRi9GMi1GIzYmLUY2Ni1RKiZ1bWludXMwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZcby1JI21uR0YkNiRRIjFGJ0Y5Ri9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGOUY5LUYjNiUtRiw2JlEiVEYnLyUlc2l6ZUdRIzEwRicvRjBGPUY5Ri9GMkZiby1GUzYmLUYjNigtSSVtc3ViR0YkNiUtRiw2J1EidkYnLyUlYm9sZEdGMUZdcC9GM1ElYm9sZEYnLyUrZm9udHdlaWdodEdGW3EtRiM2JUZeb0YvRjIvJS9zdWJzY3JpcHRzaGlmdEdGZG8tRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTlEsMC4zMzMzMzMzZW1GJy1GNjYtUSkmaGVsbGlwO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xNjY2NjY3ZW1GJy9GTkZeckZicS1GY3A2JUZlcC1GIzYlLUYsNiVRIm5GJ0YvRjJGL0YyRmBxRjlGOS8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Y5.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain the Cholesky decomposition of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= and the factor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:
Enter the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=Context Menu: Solvers and Forms\342\211\273CholeskyContext Menu: Assign to a Name\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiTEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=SSJCRzYiprint(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVE3Q2hvbGVza3l+ZGVjb21wb3NpdGlvbkYnRi9GMkZDLinearAlgebra:-LUDecomposition( , 'method=Cholesky' );LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExYXNzaWdufnRvfmF+bmFtZUYnRi9GMkZDL := : 'L';
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain the matrix 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:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:
Obtain the matrix 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Context Menu: Assign to a Name\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkKSokSSJMR0YoISIiSSMlVEdGKEkiTUdGKA==print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExYXNzaWdufnRvfmF+bmFtZUYnRi9GMkZDS := : 'S';
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Verify the simultaneous diagonalizations 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 and 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:
Enter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiZRIlRGJy8lJXNpemVHUSMxMEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiVRIkFGJ0YyRjUtSSNtb0dGJDYtUSJ+RidGQi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhuRi5GQg==
Context Menu: Evaluate and Display InlineLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkLUYjNiQpSSJTR0YoSSMlVEdGKEkiQUdGKEYw = print(); # input placeholderJSFHEnter LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiZRIlRGJy8lJXNpemVHUSMxMEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiVRIkJGJ0YyRjUtSSNtb0dGJDYtUSJ+RidGQi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhuRi5GQg==
Context Menu: Evaluate and Display InlineLUkwZGVsYXlEb3RQcm9kdWN0RzYkJSpwcm90ZWN0ZWRHL0krbW9kdWxlbmFtZUc2IkksVHlwZXNldHRpbmdHNiRGJUkoX3N5c2xpYkdGKDYkLUYjNiQpSSJTR0YoSSMlVEdGKEkiQkdGKEYw = print(); # input placeholderJSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=The order in which Maple returns the eigenpairs of the matrix 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 determines which of the four possible forms for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= these computations will produce. That there are four such possibilities can be established from first principles, as shown in the following section.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">Simultaneous Diagonalization from First Principles</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= as the 2 \303\227 2 identity matrix. In order to use the symbol LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= for the identity matrix, its connection to the imaginary unit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGOkY6RjpGK0Y6 must be broken. This can be done with the commandQyQtSSppbnRlcmZhY2VHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy9JLmltYWdpbmFyeXVuaXRHRihJImlHRighIiI=or with the following task template.
Tools\342\211\273Tasks\342\211\273Browse: Algebra\342\211\273Complex Arithmetic\342\211\273Set Imaginary Unit
Notation for Imaginary Unit LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JUYrLUkmbXNxcnRHRiQ2Iy1GIzYlLUkjbW9HRiQ2LVEqJnVtaW51czA7RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y/LyUpc3RyZXRjaHlHRj8vJSpzeW1tZXRyaWNHRj8vJShsYXJnZW9wR0Y/LyUubW92YWJsZWxpbWl0c0dGPy8lJ2FjY2VudEdGPy8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRk4tSSNtbkdGJDYkUSIxRidGOkY6RjpGK0Y6use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
module ()
local q;
Do(%RadioButton2(visible)=true);
Do(%RadioButton3(visible)=true);
Do(%RadioButton4(visible)=true);
Do(%Button1(visible)=true);
Do(%Button2(visible)=true);
Do(%MathContainer0(visible)=true);
Do(%Label0(visible)=true);
Do(%Label1(visible)=true);
Do(%Label2(visible)=true);
q := interface(imaginaryunit);
if q=I then Do(%RadioButton2 = true);Do(%MathContainer0 = NULL);
elif q=i then Do(%RadioButton3 = true);Do(%MathContainer0 = NULL);
elif q=j then Do(%RadioButton4 = true);Do(%MathContainer0 = NULL);
else Do(%MathContainer0 = MathML[Export](convert(q,string)));
Do(%RadioButton1 = true);
end if;
end module;
end use;
JSFHuse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%RadioButton2) then interface(imaginaryunit=I);
Do(%MathContainer0 = NULL);
end if;
end use;
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JSFHuse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%RadioButton3) then interface(imaginaryunit=i);
Do(%MathContainer0 = NULL);
end if;
end use;
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JSFHuse DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
if Do(%RadioButton4) then interface(imaginaryunit=j);
Do(%MathContainer0 = NULL);
end if;
end use;
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 DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
module()
local q;
Do(q = %MathContainer0);
if nops([q])=0 then return; end if;
if not type(q,name) then return;
else interface(imaginaryunit=q);
Do(%RadioButton1(value)=true);
end if;
end module;
end use;
use DocumentTools in
# Enter Maple commands to be executed when the specified
# action is carried out on the component.
# Use:
# Do( %component_name );
# and
# Do( %component_name = value );
# to set and get properties of the component.
# You can also use arbitrary expressions
# involving components, e.g.:
# Do( %target = %input1 + 2*%input2 );
# Note the %-prefix to each component name.
# See ?CustomizingComponents for more information.
Do(%RadioButton2(visible)=false);
Do(%RadioButton3(visible)=false);
Do(%RadioButton4(visible)=false);
Do(%Label0(visible)=false);
Do(%Label1(visible)=false);
Do(%Label2(visible)=false);
Do(%Button1(visible)=false);
Do(%MathContainer0(visible)=false);
Do(%Button2(visible)=false);
end use;
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Define matrices LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy, and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:
Matrix palette: 2 \303\227 2 matrix templateContext Menu: Assign to a Name\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKlNbIj08LUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExYXNzaWdufnRvfmF+bmFtZUYnRi9GMkZDS := : 'S';Matrix palette: 2 \303\227 2 matrix templateContext Menu: Assign to a Name\342\211\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 := : 'Lambda';Matrix palette: 2 \303\227 2 identity matrixContext Menu: Assign to a Name\342\211\273LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=LUknTWF0cml4RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiMvSSQlaWRHRiciKjdBenEiLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVExYXNzaWdufnRvfmF+bmFtZUYnRi9GMkZDI := : 'I';
Equate corresponding components in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiZRIlRGJy8lJXNpemVHUSMxMEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiVRIkFGJ0YyRjUtSSNtb0dGJDYtUSJ+RidGQi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhuRi5GQg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEnJiM5MjM7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJ0Yy:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Equate corresponding components in LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYlLUYvNiZRIlRGJy8lJXNpemVHUSMxMEYnL0YzUSZmYWxzZUYnL0Y2USdub3JtYWxGJ0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUYvNiVRIkJGJ0YyRjUtSSNtb0dGJDYtUSJ+RidGQi8lJmZlbmNlR0ZBLyUqc2VwYXJhdG9yR0ZBLyUpc3RyZXRjaHlHRkEvJSpzeW1tZXRyaWNHRkEvJShsYXJnZW9wR0ZBLyUubW92YWJsZWxpbWl0c0dGQS8lJ2FjY2VudEdGQS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRmhuRi5GQg== and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiSUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Join the two sets of equations and solve the eight equations for the four unknowns LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbWlHRiQ2JVEiYUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRiw2JVEiYkYnRi9GMkY1LUYsNiVRImNGJ0YvRjJGNS1GLDYlUSJkRidGL0YyRjk=:
Reference the sets by their equation labels and form a set union with the union operator selected from the Common Symbols palette.Context Menu: Solve\342\211\273Solve (explicit)LUkmdW5pb25HJSpwcm90ZWN0ZWRHNiQ8Ji8sJiomLCZJImFHNiIiIihJImNHRiwiIiJGL0YrRi9GLyomLCZGK0YvRi4hIidGL0YuRi9GLyMiI1YiIzUvLCYqJkYqRi9JImJHRixGL0YvKiZGMUYvSSJkR0YsRi9GLyIiIS8sJiomLCZGOUYtRjtGL0YvRitGL0YvKiYsJkY5Ri9GO0YyRi9GLkYvRi9GPC8sJiomRkBGL0Y5Ri9GLyomRkJGL0Y7Ri9GLyEiIjwmLywmKiYsJkYrIiIjRi5GTUYvRitGL0YvKiYsJkYrRk1GLkYtRi9GLkYvRi9GLy8sJiomRkxGL0Y5Ri9GLyomRk9GL0Y7Ri9GL0Y8LywmKiYsJkY5Rk1GO0ZNRi9GK0YvRi8qJiwmRjlGTUY7Ri1GL0YuRi9GL0Y8LywmKiZGV0YvRjlGL0YvKiZGWUYvRjtGL0YvRi8=print(); # input placeholderLUkmbW92ZXJHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2JS1JI21vR0YkNi5RJyZyYXJyO0YnLyUwZm9udF9zdHlsZV9uYW1lR1EoMkR+TWF0aEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R1EldHJ1ZUYnLyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZHLUkmbXRleHRHRiQ2JVEmc29sdmVGJ0YvRjJGQw==solve( , Explicit=true );JSFH
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Obtain the four possible solutions for the matrix LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=:
Expression palette: Evaluation template
Reference the first solution.Context Menu: Evaluate and Display InlineLUklZXZhbEclKnByb3RlY3RlZEc2JEkiU0c2IiY2JjwmL0kiYUdGJywkKiQiJEkmIyIiIiIiIyMiIz5GLy9JImJHRicsJCokIiNgRjAjRjFGOS9JImNHRicsJEYuIyEiIyIkbCMvSSJkR0YnLCRGOCMhIiRGOTwmL0YsLCRGLiMhIz5GL0Y1L0Y8LCRGLiNGMkZARkE8JkYrL0Y2LCRGOCMhIiJGOUY7L0ZCLCRGOCMiIiRGOTwmRkdGT0ZLRlM2I0Yx = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Expression palette: Evaluation template
Reference the second solution.Context Menu: Evaluate and Display InlineLUklZXZhbEclKnByb3RlY3RlZEc2JEkiU0c2IiY2JjwmL0kiYUdGJywkKiQiJEkmIyIiIiIiIyMiIz5GLy9JImJHRicsJCokIiNgRjAjRjFGOS9JImNHRicsJEYuIyEiIyIkbCMvSSJkR0YnLCRGOCMhIiRGOTwmL0YsLCRGLiMhIz5GL0Y1L0Y8LCRGLiNGMkZARkE8JkYrL0Y2LCRGOCMhIiJGOUY7L0ZCLCRGOCMiIiRGOTwmRkdGT0ZLRlM2I0Yy = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Expression palette: Evaluation template
Reference the third solution.Context Menu: Evaluate and Display InlineLUklZXZhbEclKnByb3RlY3RlZEc2JEkiU0c2IiY2JjwmL0kiYUdGJywkKiQiJEkmIyIiIiIiIyMiIz5GLy9JImJHRicsJCokIiNgRjAjRjFGOS9JImNHRicsJEYuIyEiIyIkbCMvSSJkR0YnLCRGOCMhIiRGOTwmL0YsLCRGLiMhIz5GL0Y1L0Y8LCRGLiNGMkZARkE8JkYrL0Y2LCRGOCMhIiJGOUY7L0ZCLCRGOCMiIiRGOTwmRkdGT0ZLRlM2I0ZW = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Expression palette: Evaluation template
Reference the fourth solution.Context Menu: Evaluate and Display InlineLUklZXZhbEclKnByb3RlY3RlZEc2JEkiU0c2IiY2JjwmL0kiYUdGJywkKiQiJEkmIyIiIiIiIyMiIz5GLy9JImJHRicsJCokIiNgRjAjRjFGOS9JImNHRicsJEYuIyEiIyIkbCMvSSJkR0YnLCRGOCMhIiRGOTwmL0YsLCRGLiMhIz5GL0Y1L0Y8LCRGLiNGMkZARkE8JkYrL0Y2LCRGOCMhIiJGOUY7L0ZCLCRGOCMiIiRGOTwmRkdGT0ZLRlM2IyIiJQ== = print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Careful inspection of these four solutions shows that two effects are at play. The order of the eigenvectors is the first influence, and the orientation of the unit eigenvectors is the other. Since each agent can be in two states, there are then four possible solutions for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEiU0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic=.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=
<Text-field style="Heading 1" layout="Heading 1">References</Text-field>LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Numerical Analysis: A Practical Approach, 3rd ed., Melvin J. Marin and Robert J. Lopez, Wadsworth Publishing Company, 1991.Advanced Engineering Mathematics, Robert J. Lopez, Addison-Wesley Publishing Company, 2001. Now available from Maplesoft as the ebook Advanced Engineering Mathematics with Maple.Applied Linear Algebra, 3rd ed., Ben Noble and James W. Daniel, Prentice-Hall Publishing Company, 1988.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHJSFHTTdSMApJNVJUQUJMRV9TQVZFLzg2MjYzNDI0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIigiIiJGJyEiJ0YlTTdSMApJNVJUQUJMRV9TQVZFLzg2MjYzNDg4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiNGJkYmIiIoRiU=TTdSMApJNVJUQUJMRV9TQVZFLzg2MjYzNDg4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiNGJkYmIiIoRiU=TTdSMApJNVJUQUJMRV9TQVZFLzg2MjY1MDg4WColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjLCYjIiIqIiIjIiIiKiQiI1QjRipGKUYtLCZGJ0YqRisjISIiRilGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg2MjY1MTUyWColKWFueXRoaW5nRzYiRiVbZ2whIyUhISEiIyIjJCImO3EoISIlJCImJSlIIkYoRiU=TTdSMApJNVJUQUJMRV9TQVZFLzg2MjM4MjMyWColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiIyEiIiMiI1YiIzVGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg2MjM3MjU2WCwlKmFsZ2VicmFpY0c2IjYiW2dsISIlISEhIyUiIyIjIyEiIiIiJCIiIiMhIz4iIiVGKkYlTTdSMApJNVJUQUJMRV9TQVZFLzg2MjM4NDI0WColKmFsZ2VicmFpY0c2IjYiW2dsISMlISEhIiMiIyMiI1YiIzUhIiJGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg2MjM4NjI0WCwlKmFsZ2VicmFpY0c2IjYiW2dsISIlISEhIyUiIyIjIyEjPiIiJSIiIiMhIiIiIiRGKkYlTTdSMApJNVJUQUJMRV9TQVZFLzg2MjY1NzI4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJiIiKCIiIiUnbGFtYmRhRyEiIywmRihGKEYpRipGKywmISInRihGKSEiKEYlTTdSMApJNVJUQUJMRV9TQVZFLzg2MjY2MzA0WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIioiIiRGJyIiIkYlTTdSMApJNlJUQUJMRV9TQVZFLzEwNjIwODU5NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIiIiIiQiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEwNjIyMjE2NFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywmJSJhRyIiKiUiYkciIiQsJkYnRipGKSIiIkYlTTdSMApJNlJUQUJMRV9TQVZFLzExMzE5Nzg2MFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywkJSJiRyMhIiIiIiRGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzExMzE5OTA3NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIiIiIiQiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEwNjIwOTA0NFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyEiKSIiJiMhI1FGKEYpIyEkaCQiIzVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEwNjIxMTQ3NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIz4iIiUiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzExMzE5OTcxNlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywmJSJhRyMhIikiIiYlImJHIyEjUUYqLCZGJ0YsRisjISRoJCIjNUYlTTdSMApJNlJUQUJMRV9TQVZFLzExMzIwMjcyNFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywkJSJiRyMhIz4iIiVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzExMzIwMzk0MFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIz4iIiUiIiJGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg2MjYzNDg4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiNGJkYmIiIoRiU=TTdSMApJNlJUQUJMRV9TQVZFLzExMzIwNjc1NlgsJSlhbnl0aGluZ0c2IyYlK3RyaWFuZ3VsYXJHNiMlJmxvd2VyRzYiW2dsISIpISEhIyQiIyIjKiQiIiMjIiIiRixGKyokIiImRi1GKg==TTdSMApJNlJUQUJMRV9TQVZFLzExMzIwOTUwOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIiKCIiIywkKiZGKCMiIiJGKCIiJkYrIyEiJEYtRikjISIiRi1GJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg0MzY1MzcyWCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJCooIiQxIiMiIiIiIiNGK0YpIiImRikjISIkRigsJCokRihGKSNGKyIjYCwkKigiJGwjRilGK0YpRixGKSNGK0Y1LCQqJEY1RikjIiIkRjJGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg0OTMzNDI4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJComIiQxIiMiIiIiIiMiIiZGKSMhIz4iJEkmLCRGJyNGKyIkbCMsJComRjJGKUYsRikjISIiRjIsJEY0IyIiJEYyRiU=TTdSMApJNVJUQUJMRV9TQVZFLzg2Mjc2NjI4WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMjIiNWIiM1IiIhRikhIiJGJQ==TTdSMApJNVJUQUJMRV9TQVZFLzg2MjU1NDU2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMiIiIiIiFGJ0YmRiU=TTdSMApJNVJUQUJMRV9TQVZFLzk5NDc5MzM2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiRGKUYsJkYnNiRGLEYsRiU=TTdSMApJNlJUQUJMRV9TQVZFLzExOTE5NjY0MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjJSJhRyUiY0clImJHJSJkR0YlTTdSMApJNVJUQUJMRV9TQVZFLzk5NTM0MjU2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMmJSJtRzYkIiIiRikmRic2JCIiI0YpJkYnNiRGKUYsJkYnNiRGLEYsRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEwODUyMTA2NFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIjViIjNSIiIUYpISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEwMDMyMTI5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNlJUQUJMRV9TQVZFLzEwMDMyMTI5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNVJUQUJMRV9TQVZFLzk5NzA5ODU2WCwlKWFueXRoaW5nRzYiRiVbZ2whIiUhISEjJSIjIiMsJiomLCYlImFHIiIoJSJjRyIiIkYsRilGLEYsKiYsJkYpRixGKyEiJ0YsRitGLEYsLCYqJiwmJSJiR0YqJSJkR0YsRixGKUYsRiwqJiwmRjNGLEY0Ri9GLEYrRixGLCwmKiZGKEYsRjNGLEYsKiZGLkYsRjRGLEYsLCYqJkYyRixGM0YsRiwqJkY2RixGNEYsRixGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEwODUyMTA2NFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIjViIjNSIiIUYpISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEwMDU3MzE5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCYqJiwmJSJhRyIiIyUiY0dGKiIiIkYpRixGLComLCZGKUYqRisiIihGLEYrRixGLCwmKiYsJiUiYkdGKiUiZEdGKkYsRilGLEYsKiYsJkYzRipGNEYvRixGK0YsRiwsJiomRihGLEYzRixGLComRi5GLEY0RixGLCwmKiZGMkYsRjNGLEYsKiZGNkYsRjRGLEYsRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEwMDMyMTI5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNlJUQUJMRV9TQVZFLzEwNjMyNTQyMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyIjPkYoLCRGJyMhIiMiJGwjLCQqJCIjYEYpI0YqRjQsJEYzIyEiJEY0RiU=TTdSMApJNlJUQUJMRV9TQVZFLzEwNjkyNTU0OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyEjPkYoLCRGJyNGKyIkbCMsJCokIiNgRikjRipGMywkRjIjISIkRjNGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEwNjkxNTAyMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyIjPkYoLCRGJyMhIiMiJGwjLCQqJCIjYEYpIyEiIkY0LCRGMyMiIiRGNEYlTTdSMApJNlJUQUJMRV9TQVZFLzEwNzQ3ODEwOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyEjPkYoLCRGJyNGKyIkbCMsJCokIiNgRikjISIiRjMsJEYyIyIiJEYzRiU=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=JSFHTTdSMApJNlJUQUJMRV9TQVZFLzE0ODM4OTIyOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIoIiIiRichIidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2MjUzNzEwOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIjRiZGJiIiKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE0ODM4OTIyOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIoIiIiRichIidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MTUzMlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIoIiIiRichIidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MTU5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIjRiZGJiIiKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MTU5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIjRiZGJiIiKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MzEzMlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywmIyIiKiIiIyIiIiokIiNUI0YqRilGLSwmRidGKkYrIyEiIkYpRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MzE5NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyQiJjtxKCEiJSQiJiUpSCJGKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5ODgyOFgqJSphbGdlYnJhaWNHNiJGJVtnbCEjJSEhISIjIiMjIiNWIiM1ISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5ODg5MlgsJSphbGdlYnJhaWNHNiJGJVtnbCEiJSEhISMlIiMiIyMhIz4iIiUiIiIjISIiIiIkRilGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwNDgyMFgqJSphbGdlYnJhaWNHNiJGJVtnbCEjJSEhISIjIiMhIiIjIiNWIiM1RiU=TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwNDg4NFgsJSphbGdlYnJhaWNHNiJGJVtnbCEiJSEhISMlIiMiIyMhIiIiIiQiIiIjISM+IiIlRilGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwNTU4OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCYiIigiIiIlJ2xhbWJkYUchIiMsJkYoRihGKUYqRissJiEiJ0YoRikhIihGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwNjEwMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIqIiIkRiciIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwODE0OFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIiIiIiQiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwODg1MlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywmJSJhRyIiKiUiYkciIiQsJkYnRipGKSIiIkYlTTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwOTU1NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywkJSJiRyMhIiIiIiRGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzE2NjUwOTYyMFgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIiIiIiQiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMDAwNFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyEiKSIiJiMhI1FGKEYpIyEkaCQiIzVGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMjA1MlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIz4iIiUiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMjY5MlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywmJSJhRyMhIikiIiYlImJHIyEjUUYqLCZGJ0YsRisjISRoJCIjNUYlTTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMzAxMlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIywkJSJiRyMhIz4iIiVGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMzA3NlgqJSlhbnl0aGluZ0c2IkYlW2dsISMlISEhIiMiIyMhIz4iIiUiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE0ODM5MTU5NlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIjRiZGJiIiKEYlTTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxMzM5NlgsJSlhbnl0aGluZ0c2IyYlK3RyaWFuZ3VsYXJHNiMlJmxvd2VyRzYiW2dsISIpISEhIyQiIyIjKiQiIiMjIiIiRixGKyokIiImRi1GKg==TTdSMApJNlJUQUJMRV9TQVZFLzE2NjUxNDc0MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIiKCIiIywkKiZGKCMiIiJGKCIiJkYrIyEiJEYtRikjISIiRi1GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc4OTc4MFgqJSphbGdlYnJhaWNHNiJGJVtnbCEjJSEhISIjIiMjIiNWIiM1ISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc4OTg0NFgsJSphbGdlYnJhaWNHNiJGJVtnbCEiJSEhISMlIiMiIywkKiYiIiMjIiIiRigiIiZGKSMhIiQiIiVGKiwkRicjRigiIzpGKkYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MDc4OTg0NFgsJSphbGdlYnJhaWNHNiJGJVtnbCEiJSEhISMlIiMiIywkKiYiIiMjIiIiRigiIiZGKSMhIiQiIiVGKiwkRicjRigiIzpGKkYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MDE2NFgqJSphbGdlYnJhaWNHNiJGJVtnbCEjJSEhISIjIiMsJComIiIjIyIiIkYoIiImRikjISIkIiIlRipGJQ==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TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MTI1MlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJiIkMSIjIiIiIiIjIiImRikjISM+IiRJJiwkRicjRisiJGwjLCQqJkYyRilGLEYpIyEiIkYyLCRGNCMiIiRGMkYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MTcwMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIjViIjNSIiIUYpISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MjE0OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MTgxNDg0MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjJSJhRyUiY0clImJHJSJkR0YlTTdSMApJNlJUQUJMRV9TQVZFLzE3MjA2MjM0OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIjViIjNSIiIUYpISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MjIxMlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MjY2MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCYqJiwmJSJhRyIiKCUiY0ciIiJGLEYpRixGLComLCZGKUYsRishIidGLEYrRixGLCwmKiYsJiUiYkdGKiUiZEdGLEYsRilGLEYsKiYsJkYzRixGNEYvRixGK0YsRiwsJiomRihGLEYzRixGLComRi5GLEY0RixGLCwmKiZGMkYsRjNGLEYsKiZGNkYsRjRGLEYsRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MjA2MjM0OFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIyIjViIjNSIiIUYpISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MzEwOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCYqJiwmJSJhRyIiIyUiY0dGKiIiIkYpRixGLComLCZGKUYqRisiIihGLEYrRixGLCwmKiYsJiUiYkdGKiUiZEdGKkYsRilGLEYsKiYsJkYzRipGNEYvRixGK0YsRiwsJiomRihGLEYzRixGLComRi5GLEY0RixGLCwmKiZGMkYsRjNGLEYsKiZGNkYsRjRGLEYsRiU=TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc5MjIxMlgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjIiIiIiIhRidGJkYlTTdSMApJNlJUQUJMRV9TQVZFLzI0NDE0MTgyOFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyIjPkYoLCRGJyMhIiMiJGwjLCQqJCIjYEYpI0YqRjQsJEYzIyEiJEY0RiU=TTdSMApJNlJUQUJMRV9TQVZFLzE2MzUxMDk4MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyEjPkYoLCRGJyNGKyIkbCMsJCokIiNgRikjRipGMywkRjIjISIkRjNGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzE3MDc4MDEyMFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyIjPkYoLCRGJyMhIiMiJGwjLCQqJCIjYEYpIyEiIkY0LCRGMyMiIiRGNEYlTTdSMApJNlJUQUJMRV9TQVZFLzI0NTE1ODM4MFgsJSlhbnl0aGluZ0c2IkYlW2dsISIlISEhIyUiIyIjLCQqJCIkSSYjIiIiIiIjIyEjPkYoLCRGJyNGKyIkbCMsJCokIiNgRikjISIiRjMsJEYyIyIiJEYzRiU=