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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!

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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 $m_B, m_C | m_A$.

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

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