Functions of a Square Matrix

Let $f(\lambda )={\beta }_0+{\beta }_1\lambda +\cdots +{\beta }_n{\lambda }^n$ be a polynomial. Then if the input is a matrix, $f(A)={\beta }_0\mathbb{I}+{\beta }_{1}A+\cdots +{\beta }_n{A}^n$.

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Property 1: Functions of similar matrices

$$f(PA{P}^{-1})=Pf(A)P^{-1}$$

Property 2: Functions of block diagonal matrices

$$A=\left[\begin{array}{cccc}{A}_1& & & \\ & A_2& & \\ & & a_3& \\ & & & a_4\end{array}\right]\to f(A)=\left[\begin{array}{cccc}f(A_1)& & & \\ & f(A_2)& & \\ & & f(a_3)& \\ & & & f(a_4)\end{array}\right]$$

Both of these properties are true for functions of $A$ in general, not just polynomials.

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For example, let’s say you want to compute

$$A=\left[\begin{array}{cc}2& 4\\ 0& 1\end{array}\right],f(A)=A^{10}$$

1. Find the eigenvalues of $A$
   • ${\lambda }_1=2,{\lambda }_2=1$
2. Solve the set of equations $h(\lambda )={\beta }_0+{\beta }_1\lambda$, $f(\lambda )={\lambda }^{10}$
   • ${\beta }_0=-1022$, and ${\beta }_1=1023$
3. Solve $f(A)={\beta }_0\mathbb{I}+{\beta }_{1}A$
   • $f(A)=\left[\begin{array}{cc}1024& 4092\\ 0& 1\end{array}\right]=A^{10}$