Block matrix - Knowlpedia

A block matrix is a matrix written as an array of rectangular submatrices after partitioning its row and column indices into groups. For example,

M=\begin{pmatrix}A&B\\ C&D\end{pmatrix}.

Compatible block partitions permit the usual addition and multiplication rules, with matrix products in place of scalar products. Blocks on the block diagonal, such as $A,D$ , are diagonal blocks; they need not themselves be diagonal matrices.

Decoupling

If all off-diagonal blocks vanish, the matrix is block diagonal. Its action preserves the corresponding coordinate subspaces, and a differential system $y'=My$ splits into those blocks. A small off-diagonal block instead describes coupling; it is not an exact invariant decomposition.