The MatrixConvergence command determines whether the square matrix A is convergent, in the sense that $\underset{k\to \mathrm{\infty }}{lim}{\left(A^k\right)}_{i,j}=0$ 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.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{NumericalAnalysis}\right]\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[\frac{1}{2}\,\frac{1}{4}\,\frac{1}{5}\right]\,\left[\frac{1}{3}\,\frac{1}{2}\,\frac{1}{6}\right]\,\left[\frac{1}{6}\,\frac{1}{7}\,\frac{1}{3}\right]\right]\right)$

$A≔\left[\begin{array}{ccc}\frac{1}{2}& \frac{1}{4}& \frac{1}{5}\\ \frac{1}{3}& \frac{1}{2}& \frac{1}{6}\\ \frac{1}{6}& \frac{1}{7}& \frac{1}{3}\end{array}\right]$

$\mathrm{MatrixConvergence}\left(A\right)$

$\mathrm{true}$