I found somewhere in a book, that it is defined the polynomial $p\in\mathbb C[X]$ with k degree and then they write down the polynomial for 2 values: $A\in M_n$ and $a\in\mathbb C$, like this:
$p(A)=a_0*I_n+a_1*a+...+a_k*A^k$ $p(a)=a_0+a_1*a+...+a_k*a^k$ What I don't get is why the coefficient of 0-degreed indeterminate change ( from a_0*I_n to a_0*1), or is it considered a s change? If not, then how should see this transformation? Perhaps it has something to do with the generalized way $X^0$ is viewed in a polynomial, where $X$ is indeterminate? If so, then why is it considered to be correct? Thanks in advance!
Given a polynomial $$p(x)=a_nx^n+\ldots+a_1x+a_0$$ for a real variable $x$ the value of the polynomial $p$ at a matrix $A$ is defined as $$p(A)=a_nA^n+\ldots+a_1A+a_0I_n.$$ Otherwise, you cannot add number $a_0$ to matrices. This is just how it is defined.