I am currently struggling with an exercise where I have to treat a Block diagonal matrix (so it is a square matrix, where square block matrices are down the diagonal). Now I was wondering whether we can say something about the characteristic or minimal polynomial of the whole matrix, if we know the characteristic and minimal polynomial of the blocks? Anything you know could be helpful!
Linear Algebra – Minpoly and Charpoly of Block Diagonal Matrix
block matricescharacteristic polynomiallinear algebramatricesminimal-polynomials
Best Answer
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.