25 Functions of a matrix

Definition As in MATH220, we form polynomials in $A$. Let $I$ be the $n\times n$ identity matrix. We can form $A,A^2,A^3,\mathrm{\dots }$ by matrix multiplication, and hence given $g(z)=a_m{z}^m+\mathrm{\dots }+a_0$ we build polynomials

for complex coefficients $a_j$. We regard $A^0=I$, the identity matrix.