Minpoly and Charpoly of block diagonal matrix

The characteristic polynomial of the block matrix $A$ is just the product of the characteristic polynomials of the blocks, just remember that the determinant of a block diagonal matrix is the product of the determinants of the blocks, and apply this to the block diagonal matrix $x I - A$.

The minimal polynomial is slightly subtler, as it is the $\operatorname{lcm}$ of the minimal polynomials of the blocks.

To prove this, notice first that the minimal polynomial $m(x)$ of $A$ vanishes when computed on each block, so the minimal polynomial $m_i(x)$ of the $i$-th block divides $m(x)$. So $\operatorname{lcm}(m_i(x))$ divides $m(x)$, but it's easy to see that $\operatorname{lcm}(m_i(x))$ annihilates each block, so $\operatorname{lcm}(m_i(x)) = m(x)$.

Thanks to @Lipschitz for the correction in the comment below.

Tags:

Linear Algebra

Matrices

Block Matrices

Minimal Polynomials

Characteristic Polynomial

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