Estimating matrix raised to a power

Question

What would be the best and easiest way to estimate a matrix to a power?

Specifically, let's say you have matrix A:

$$A = \left[ \begin{matrix} 3/4 & 1/4 & 0 & 0 \\\ 1/4 & 1/2 & 1/4 & 0 \\\ 0 & 1/4 & 1/2 & 1/4 \\\ 0 & 0 & 1/4 & 3/4 \end{matrix} \right]$$

How can I estimate $A^{100}$ without using any calculators? How can I use the spectral decomposition theorem for symmetric matrices and show how it gives an approximate estimation?

Would greatly appreciate any help, thank you!

Answer 1

Since $A$ is real symmetric, $A$ is diagonalizable over $\mathbb{R}$. $1$ is an eigenvalue of $A$ and $\ker(A-I)$ admits $u=[1/2,1/2,1/2,1/2]^T$ as a unitary basis. Since $A$ is stochastic, $1$ is semi simple and, therefore, has multiplicity $1$.

By Gerschgorin, $1$ is the unique eigenvalue of maximal modulus $1$ and the $3$ other eigenvalues $a,b,c$ are in $[0,1[$.

There is $P=[u,?,?,?]\in O(4)$ s.t. $A=Pdiag(1,a,b,c)P^T$ and $A^{100}=Pdiag(1,a^{100},b^{100},c^{100})P^T$.

Finally $A^{100}\approx Pdiag(1,0,0,0)P^T=uu^T=\dfrac{1}{4}U$ where $U$ is the matrix of ones.