\[A\]
. The limiting form of high powers of \[A\]
depends on the nature of the eigenvalues. Where there is a single real dominant eigenvalue \[\lambda\]
the ratio of elements of powers of \[A\]
tends to that eigenvalue: \[lim_{m \rightarrow \infty} \frac{a^{m+1}_{ij}}{a^m_{ij}}= \lambda\]
(1)This means that the roots of the equation
\[a^m_{ij} \lambda^2-2 a^{m+1}_{ij} \lambda +a^{m+2}_{ij}=0\]
get closer the the eigenvalue \[\lambda\]
of \[A\]
.Example: Let
\[A=\left( \begin{array}{ccc} 4 & 1 & 2 \\ 2 & 4 & -3 \\ 3 & 1 & 3 \end{array} \right)\]
.\[A^8=\left( \begin{array}{ccc} 952149 & 625000 & 63476 \\ -463868 & -234375 & -161132 \\ 952148 & 625000 & 63477 \end{array} \right)\]
.\[A^9=\left( \begin{array}{ccc} 5249024 & 3515625 & 219726 \\ -2807618 & -1562500 & -708007 \\ 5249023 & 3515625 & 219727 \end{array} \right)\]
.\[A^{10}=\left( \begin{array}{ccc} 28686524 & 19531250 & 610351 \\ -16479493 & -9765625 & -3051757 \\ 28586523 & 19531250 & 610352 \end{array} \right)\]
.Let
\[m=8\]
in (1) and use as \[a_{ij}\]
the element \[a_{11}\]
.We have
\[952149 \lambda^2-2 \times 5249024 \lambda +28686524=0\]
.The roots are
\[\lambda+5.000008, \: 6.025628\]
.Using
\[a_{33}\]
instead gives the equation \[62477 \lambda^2 - 2 \times 219729 \lambda +610352\]
gives \[\lambda=4.999956, \: 1.923078\]
.Hence the dominant eigenvalue is
\[\lambda=5\]
.