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Student[LinearAlgebra]

 IsSimilar
 determine if two Matrices are similar

 Calling Sequence IsSimilar(A, B, options)

Parameters

 A - square Matrix B - square Matrix options - (optional) parameters; for a complete list, see LinearAlgebra[IsSimilar]

Description

 • The IsSimilar(A, B) command returns false if A and B are not similar, and otherwise returns the expression sequence true, P, where P is the similarity transformation matrix, that is, $A={P}^{\left(-1\right)}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}P$.
 If either of the Matrices has a floating-point data type, then each must evaluate to a purely floating-point Matrix. In this case, the result of IsSimilar is achieved by making a numeric comparison of the eigenvalues of each Matrix.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{LinearAlgebra}]\right):$
 > $\mathrm{IsSimilar}\left(⟨⟨1,2⟩|⟨3,4⟩⟩,⟨⟨9,17⟩|⟨-2,-4⟩⟩\right)$
 ${\mathrm{true}}{,}\left[\begin{array}{cc}{1}& {0}\\ {4}& {-}\frac{{3}}{{2}}\end{array}\right]$ (1)
 > $A≔⟨⟨3,1⟩|⟨2,2⟩⟩$
 ${A}{:=}\left[\begin{array}{rr}{3}& {2}\\ {1}& {2}\end{array}\right]$ (2)
 > $B≔\mathrm{DiagonalMatrix}\left(\left[1,4\right]\right)$
 ${B}{:=}\left[\begin{array}{rr}{1}& {0}\\ {0}& {4}\end{array}\right]$ (3)
 > $S≔\mathrm{IsSimilar}\left(A,B\right)$
 ${S}{:=}{\mathrm{true}}{,}\left[\begin{array}{cc}{-}\frac{{1}}{{3}}& \frac{{2}}{{3}}\\ \frac{{2}}{{3}}& \frac{{2}}{{3}}\end{array}\right]$ (4)
 > ${S}_{2}^{\left(-1\right)}\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}B\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}.\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}{S}_{2}$
 $\left[\begin{array}{rr}{3}& {2}\\ {1}& {2}\end{array}\right]$ (5)