Cayley-Hamilton and Matrix Similarity

For computing the nth power of a square matrix A

This bit of linear algebra is a useful tool in statistics because it allows one to easily compute the nth exponent of a full-rank square matrix A. It rests on two foundational concepts: Cayley-Mailton theorem and Matrix Similarity.

Cayley-Hamilton theory states:

Matrix Similarity states:

we should also note that A given square matrix A is always similar to itself.

The Method

1. Use Cayley-Hamilton equation to find all eigenvalues $\lambda_{1},\lambda_{2},etc.$

2. By the eigenvalue property find their corresponding n-dimensional eigenvectors $\textbf{X}_{1},\textbf{X}_{2},etc.$ through the system of equations given by $\textbf{A}\textbf{X}_{n}=\lambda_{n}\textbf{X}_{n}$.

3. Assemble $n \times n$ matrix $\textbf{P}=[\textbf{X}_{1},\textbf{X}_{2},etc.]$

4. Assemble $n \times n$ matrix $\textbf{D}=\begin{bmatrix} \lambda_{1} & 0 & 0 \\ 0 & \lambda_{2} & 0 \\ 0 & 0 & etc \end{bmatrix}$

5. See that $\textbf{A}$ and $\textbf{D}$ are similar to each other through $\textbf{P}$. Thus: $\textbf{A}^{n}=\textbf{P}*\textbf{D}^{n}*\textbf{P}^{-1}$