Matrix functions

Do not confuse with Matrices of functions.

Polynomials of functions

Let $A$ be matrix and $p(x)=x^3+2x+2$. Then

$$p(A)=A^3+2A+2I$$

Matrix exponential

Recall the Taylor expansion of the exponential function $e^x={\sum }_{n=0}^{\infty }\frac{x^n}{n!}$.

Let $A$ be a $n\times n$ matrix. Then

$$e^{At}=\sum _{n=0}^{\infty }\frac{A^n{t}^n}{n!}=I+At+\frac{A^2{t}^2}{2}+\frac{A^3{t}^3}{2\times 3}\dots$$

Any constant matrix $A$ makes the above function convergent. Therefore $e^{At}$ is defined for any constant matrix $A$.