Matrix polynomial
hint:
\( P(x)= \sum_{i=0}^{n}a_ix^i= a_0x^0+a_1x^1+\dots+a_nx^n \)
Definition
\( P(A)=\sum_{i=0}^{n}a_iA^i= a_0I+a_1A^1+a_2A^2+\dots+a_nA^n \)
Suppose there are two polynomials, \( f(x),g(x) \), \( A\in F^{n\times n} \).
\( f(A)g(A)=g(A)f(A) \)
Proof
\(\because r(x)= f(x)g(x)=g(x)f(x) \)
\(\therefore r(A)= f(A)g(A)=g(A)f(A) \)
If \( g(A) \) is a nonsingular matrix. \( f(A)g(A)^{-1}= g(A)^{-1}f(A)\)
Proof
\(f(A)g(A)=g(A)f(A) \)
\( \exists!g(A)^{-1} \)
\( f(A)=g(A)f(A)g(A)^{-1} \)
\( \Rightarrow g(A)^{-1}f(A)=f(A)g(A)^{-1} \)
\( \forall \) means all true is true.
\( \exists \) means for some true is true(one or more).
\( \exists! \) means only one true is true.
Reference
wiki
黃子嘉-線性代數