I'm trying to prove that the minimal polynomial of a diagonal block matrix i.e. a matrix
$A = \begin{bmatrix}
B & 0 \\[0.3em]
0 & C \\[0.3em]
\end{bmatrix}$
is the less common multiple of the minimal polynomials of matrix B and matrix C.
Well I checked down the answer that the user @AndreasCaranti gave here:
Minpoly and Charpoly of block diagonal matrix
However there is a crucial aspect of the proof that I would like to have some clarification please:
- "the minimal polynomial $m(x)$ of $A$ vanishes when computed on each block do the minimal polynomial $m_i(x)$ of the $i$-th block divides $m(x)$". Why? Is this a property of block matrices? Why does the minimal polynomial of each block divides the matrix.
Thanks!
Best Answer
Given a polynomial $p \in \mathbb{F}[X]$, we have
$$ p \begin{pmatrix} B & 0 \\ 0 & C \end{pmatrix} = \begin{pmatrix} p(B) & 0 \\ 0 & p(C) \end{pmatrix}. $$
Since $m_A(A) = 0$, we have $m_A(B) = 0, m_A(C) = 0$ which implies that $m_B, m_C | m_A$.