Issimilar - Maple Programming Help

Home : Support : Online Help : Mathematics : Inert Functions : Issimilar

Issimilar

determine similarity of matrices

 Calling Sequence Issimilar(A, B) Issimilar(A, B, 'P')

Parameters

 A - square Matrix B - square Matrix 'P' - (optional) assigned a transformation matrix

Description

 • The function Issimilar(A, B) is a placeholder for the boolean valued function that returns true if A is similar to B and false otherwise.  It is used in conjunction with either mod or evala.
 • If called in the form Issimilar(A, B,'P'), then P will be assigned a transformation matrix such that $A=\mathrm{inverse}\left(P\right)BP$.
 • The call Issimilar(A, B) mod p determines if A is similar to B modulo p which is a prime integer.  The entries of A and B must have rational coefficients or coefficients from an algebraic extension of the integers modular p.
 • The call evala(Issimilar(A, B)) determines if A is similar to B where the entries of A and B are algebraic numbers (or functions) defined by RootOfs.

Examples

 > $\mathrm{with}\left(\mathrm{LinearAlgebra}\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[1,2,3\right],\left[4,5,6\right],\left[7,8,9\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}{1}& {2}& {3}\\ {4}& {5}& {6}\\ {7}& {8}& {9}\end{array}\right]$ (1)
 > $p≔11$
 ${p}{≔}{11}$ (2)
 > $\mathrm{cA}≔\mathrm{CharacteristicPolynomial}\left(A,x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p$
 ${\mathrm{cA}}{≔}{{x}}^{{3}}{+}{7}{}{{x}}^{{2}}{+}{4}{}{x}$ (3)
 > $B≔\mathrm{CompanionMatrix}\left(\mathrm{cA},x\right)$
 ${B}{≔}\left[\begin{array}{ccc}{0}& {0}& {0}\\ {1}& {0}& {-4}\\ {0}& {1}& {-7}\end{array}\right]$ (4)
 > $\mathrm{Issimilar}\left(A,B,'P'\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}p$
 ${\mathrm{true}}$ (5)
 > $P$
 $\left[\begin{array}{ccc}{8}& {6}& {8}\\ {8}& {2}& {3}\\ {1}& {0}& {0}\end{array}\right]$ (6)
 > ${P}^{-1}·B·P\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{mod}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}11$
 $\left[\begin{array}{ccc}{1}& {2}& {3}\\ {4}& {5}& {6}\\ {7}& {8}& {9}\end{array}\right]$ (7)