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LinearAlgebra

 Eigenvectors
 compute the eigenvectors of a Matrix

 Calling Sequence Eigenvectors(A, C, imp, out, options, outopts)

Parameters

 A - Matrix; Matrix whose Eigenvectors are required C - (optional) Matrix; Matrix for the generalized eigenvector problem imp - (optional) BooleanOpt(implicit); specify that roots of the characteristic polynomial contain implicit RootOfs or radicals out - (optional) equation of the form output = obj where obj is one of 'values', 'vectors', or 'list', or a list containing one or more of these names; selects format of the result object options - (optional); constructor options for the result object(s) outopts - (optional) equation(s) of the form outputoptions[o]=list where o is one of 'values', 'vectors', or 'list'; constructor options for the specified result object

Description

 • The Eigenvectors(A) command solves the simple eigenvector problem by returning an expression sequence whose first member is a Vector V of the eigenvalues of A, and whose second member is a Matrix E whose columns are the eigenvectors of A.  The ith column of E is an eigenvector associated with the ith eigenvalue of the returned Vector V.
 In the simple floating-point eigenvector problem, if A has either the symmetric or the hermitian indexing function then the returned eigenvalue object has $\mathrm{float}[8]$ or $\mathrm{sfloat}$ datatype. Otherwise the returned eigenvalue object has $\mathrm{complex}[8]$ or $\mathrm{complex}\left(\mathrm{sfloat}\right)$ datatype.
 • The Eigenvectors(..) function solves the simple eigenvector problem $A·x=\mathrm{\lambda }·x$ and the generalized eigenvalue problem $A·x=\mathrm{\lambda }·C·x$.  The eigenvalues V[i] and the nonzero eigenvectors E[1 .. -1, i] are such that for the simple eigenvalue problem $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}={V}_{i}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}$, and for the generalized problem $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}={V}_{i}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}C\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{E}_{1..-1,i}$.
 • The Eigenvectors(A, C) command solves the generalized eigenvector problem.
 In the generalized floating-point eigenvector problem, if A and C have either symmetric or hermitian indexing functions and C also has the positive_definite attribute then the returned eigenvalue object has $\mathrm{float}[8]$ or $\mathrm{sfloat}$ datatype. Otherwise the returned eigenvalue object has $\mathrm{complex}[8]$ or $\mathrm{complex}\left(\mathrm{sfloat}\right)$ datatype.
 • With an eigenvalue of multiplicity $k>1$, there may be fewer than $k$ linearly independent eigenvectors. In this case, the matrix is called defective.  By design, the returned matrix always has full column dimension.  Therefore, in the defective case, some of the columns that are returned are zero.  Thus, they are not eigenvectors.  With the option, output=list, only eigenvectors are returned.  For more information, see LinearAlgebra[JordanForm] and LinearAlgebra[SchurForm].
 • The format in which the Eigenvectors of A are returned is determined by parameter out.  By default, an expression sequence is returned as described above. In the case that the output is specified as list, a list of lists is returned.  The first element of each sublist is an eigenvalue, the second element is its multiplicity, and the third element is a set of linearly independent eigenvectors corresponding to that eigenvalue.
 • The options option provides additional information (readonly, shape, storage, order, datatype, and attributes) to the Matrix or Vector constructor that builds the result(s). These options may also be provided in the form outputoptions[o]=[...], where [...] represents a Maple list.  If a constructor option is provided in both the calling sequence directly and in an outputoptions[o] option, the latter takes precedence (regardless of the order).
 • This function is part of the LinearAlgebra package, and so it can be used in the form Eigenvectors(..) only after executing the command with(LinearAlgebra). However, it can always be accessed through the long form of the command by using LinearAlgebra[Eigenvectors](..).

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$

Find the eigenvalues and eigenvectors of the Matrix $A$.

 > $A≔⟨⟨-1,-3,-6⟩|⟨3,5,6⟩|⟨-3,-3,-4⟩⟩$
 ${A}{:=}\left[\begin{array}{rrr}{-}{1}& {3}& {-}{3}\\ {-}{3}& {5}& {-}{3}\\ {-}{6}& {6}& {-}{4}\end{array}\right]$ (1)
 > $v,e≔\mathrm{Eigenvectors}\left(A\right)$
 ${v}{,}{e}{:=}\left[\begin{array}{r}{-}{4}\\ {2}\\ {2}\end{array}\right]{,}\left[\begin{array}{ccc}\frac{{1}}{{2}}& {-}{1}& {1}\\ \frac{{1}}{{2}}& {0}& {1}\\ {1}& {1}& {0}\end{array}\right]$ (2)

Verify $A·x=\mathrm{\lambda }x$ for the second eigenvalue and second eigenvector.

 > $A\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{e}_{1..-1,2}={v}_{2}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{e}_{1..-1,2}$
 $\left[\begin{array}{r}{-}{2}\\ {0}\\ {2}\end{array}\right]{=}\left[\begin{array}{r}{-}{2}\\ {0}\\ {2}\end{array}\right]$ (3)

Find the eigenvectors of the Matrix $B$ using the option output=list.

 > $B≔⟨⟨1,2,3⟩|⟨2,4,6⟩|⟨5,10,15⟩⟩$
 ${B}{:=}\left[\begin{array}{rrr}{1}& {2}& {5}\\ {2}& {4}& {10}\\ {3}& {6}& {15}\end{array}\right]$ (4)
 > $\mathrm{Eigenvectors}\left(B,\mathrm{output}='\mathrm{list}'\right)$
 $\left[\left[{0}{,}{2}{,}\left\{\left[\begin{array}{r}{-}{5}\\ {0}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{-}{2}\\ {1}\\ {0}\end{array}\right]\right\}\right]{,}\left[{20}{,}{1}{,}\left\{\left[\begin{array}{c}\frac{{1}}{{3}}\\ \frac{{2}}{{3}}\\ {1}\end{array}\right]\right\}\right]\right]$ (5)

Find the generalized eigenvectors of $M$ and $N$.

 > $M≔\mathrm{Matrix}\left(\left[\left[6.,8.,5.\right],\left[8.,8.,9.\right],\left[5.,9.,6.\right]\right],\mathrm{datatype}=\mathrm{float}\right):$
 > $N≔\mathrm{Matrix}\left(\left[\left[6.,3.,3.\right],\left[3.,8.,8.\right],\left[3.,8.,9.\right]\right],\mathrm{datatype}=\mathrm{float}\right):$
 > $v,e≔\mathrm{Eigenvectors}\left(M,N,\mathrm{output}=\left['\mathrm{vectors}','\mathrm{values}'\right]\right)$
 ${v}{,}{e}{:=}\left[\begin{array}{ccc}{-}{0.139566030823037}{+}{0.}{}{I}& {-}{0.940687706820450}{+}{0.}{}{I}& {-}{1.}{+}{0.}{}{I}\\ {1.}{+}{0.}{}{I}& {-}{1.}{+}{0.}{}{I}& {0.0726922335169216}{+}{0.}{}{I}\\ {-}{0.911459942195583}{+}{0.}{}{I}& {0.399204400339096}{+}{0.}{}{I}& {0.953350738117875}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{c}{-}{4.55651173111532}{+}{0.}{}{I}\\ {1.56423606281299}{+}{0.}{}{I}\\ {0.223044899071558}{+}{0.}{}{I}\end{array}\right]$ (6)

A simple floating-point eigenvector problem.

 > $N≔⟨⟨1.0,4.0I⟩|⟨-4.0I,1.0⟩⟩$
 ${N}{:=}\left[\begin{array}{cc}{1.0}& {-}{4.0}{}{I}\\ {4.0}{}{I}& {1.0}\end{array}\right]$ (7)
 > $\mathrm{evalsN},\mathrm{evecsN}≔\mathrm{Eigenvectors}\left(N\right)$
 ${\mathrm{evalsN}}{,}{\mathrm{evecsN}}{:=}\left[\begin{array}{c}{5.00000000000000}{+}{0.}{}{I}\\ {-}{3.}{+}{0.}{}{I}\end{array}\right]{,}\left[\begin{array}{cc}{-}{0.}{-}{0.707106781186547}{}{I}& {0.707106781186547}{+}{0.}{}{I}\\ {0.707106781186548}{+}{0.}{}{I}& {0.}{-}{0.707106781186547}{}{I}\end{array}\right]$ (8)

By default, the resulting eigenvalues are given as complex numbers.

 > $\mathrm{VectorOptions}\left(\mathrm{evalsN},\mathrm{datatype}\right),\mathrm{MatrixOptions}\left(\mathrm{evecsN},\mathrm{datatype}\right)$
 ${{\mathrm{complex}}}_{{8}}{,}{{\mathrm{complex}}}_{{8}}$ (9)

However, if $N$ has the $\mathrm{hermitian}$ indexing function, then the resulting eigenvalues are given as real floating-point numbers.

 > $\mathrm{evalsN},\mathrm{evecsN}≔\mathrm{Eigenvectors}\left(\mathrm{Matrix}\left(N,\mathrm{shape}=\mathrm{hermitian}\right)\right)$
 ${\mathrm{evalsN}}{,}{\mathrm{evecsN}}{:=}\left[\begin{array}{c}{-}{3.}\\ {5.}\end{array}\right]{,}\left[\begin{array}{cc}{0.}{-}{0.707106781186547}{}{I}& {0.}{-}{0.707106781186547}{}{I}\\ {-}{0.707106781186547}{-}{0.}{}{I}& {0.707106781186547}{+}{0.}{}{I}\end{array}\right]$ (10)
 > $\mathrm{VectorOptions}\left(\mathrm{evalsN},\mathrm{datatype}\right),\mathrm{MatrixOptions}\left(\mathrm{evecsN},\mathrm{datatype}\right)$
 ${{\mathrm{float}}}_{{8}}{,}{{\mathrm{complex}}}_{{8}}$ (11)