blocksymmetric

Block-symmetric is a term used in linear algebra to describe matrices that are partitioned into equal-sized square blocks and exhibit symmetry at the block level. Let A be partitioned into p by p blocks A_ij, each of size r by r. A is block-symmetric if, for all i and j, A_ij^T = A_ji. In particular, the diagonal blocks A_ii must be symmetric. When the underlying field is complex, the analogous condition uses the conjugate transpose, so a block-Hermitian matrix satisfies A_ij^H = A_ji.

This concept generalizes ordinary matrix symmetry. If each block is a scalar (r = 1), block-symmetry reduces to the usual condition A = A^T. The requirement that blocks are arranged in a symmetric block pattern implies that the matrix as a whole has a symmetric structure with respect to the block partition, and diagonal blocks must be symmetric (or Hermitian in the complex case).

A common example is a 2-by-2 block matrix of the form [B C; C^T D], where B and D are symmetric. More generally, A_ij and A_ji are transposes of each other, ensuring the overall matrix is symmetric with respect to the block partition.

Computationally, block-symmetric matrices enable specialized algorithms that exploit the block structure, such as block Cholesky or block LU factorizations when the matrix is also positive definite or has a suitable Schur complement. They frequently arise in discretizations of partial differential equations, finite element methods, multivariate statistics with block covariance structures, and control theory, where variables are naturally grouped into interacting blocks.