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Student[NumericalAnalysis]

 MatrixConvergence
 check whether a matrix is convergent

 Calling Sequence MatrixConvergence(A)

Parameters

 A - Matrix; a square matrix

Description

 • The MatrixConvergence command determines whether the square matrix A is convergent, in the sense that $\underset{k→\infty }{lim}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\left({A}^{k}\right)}_{i,j}=0$ for each $i,j=1..n$, where $n$ is the dimension of A.
 • A is convergent if and only if the spectral radius of A is strictly less than 1.

Examples

 > $\mathrm{with}\left(\mathrm{Student}[\mathrm{NumericalAnalysis}]\right):$
 > $A≔\mathrm{Matrix}\left(\left[\left[\frac{1}{2},\frac{1}{4},\frac{1}{5}\right],\left[\frac{1}{3},\frac{1}{2},\frac{1}{6}\right],\left[\frac{1}{6},\frac{1}{7},\frac{1}{3}\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{ccc}\frac{{1}}{{2}}& \frac{{1}}{{4}}& \frac{{1}}{{5}}\\ \frac{{1}}{{3}}& \frac{{1}}{{2}}& \frac{{1}}{{6}}\\ \frac{{1}}{{6}}& \frac{{1}}{{7}}& \frac{{1}}{{3}}\end{array}\right]$ (1)
 > $\mathrm{MatrixConvergence}\left(A\right)$
 ${\mathrm{true}}$ (2)