Block upper triangular matrix

A block upper triangular matrix is a block matrix with square diagonal blocks and zero blocks below them. Its determinant is the product of the determinants of its diagonal blocks.

If a $k$-dimensional invariant subspace is mapped to the span of the first $k$ coordinate vectors by a similarity transformation, the transformed matrix has block formIts characteristic polynomial factors as $\det (\lambda I-B)\det (\lambda I-D)$, so its eigenvalues are the eigenvalues of the two diagonal blocks, counted with algebraic multiplicity.