# Minimal polynomial of block matrix

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!

## 1 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 $m_B, m_C | m_A$.

• Hi! Just a quick question $m_C|m_A$ means $m_C$ divides $m_A$, i suppose? – Granger Obliviate Jun 13 '16 at 17:47
• @GrangerObliviate Yep. – levap Jun 13 '16 at 17:51