Block matrix

[1][2] Intuitively, a matrix interpreted as a block matrix can be visualized as the original matrix with a collection of horizontal and vertical lines, which break it up, or partition it, into a collection of smaller matrices.

Any matrix may be interpreted as a block matrix in one or more ways, with each interpretation defined by how its rows and columns are partitioned.

The original matrix is then considered as the "total" of these groups, in the sense that the

entry of the original matrix corresponds in a 1-to-1 way with some

[4] Block matrix algebra arises in general from biproducts in categories of matrices.

[5] The matrix can be visualized as divided into four blocks, as The horizontal and vertical lines have no special mathematical meaning,[6][7] but are a common way to visualize a partition.

[9] A special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed.

[11] As with the conventional trace operator, the block transpose is a linear mapping such that

, then It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors.

is carried out treating the submatrices as if they are scalars, but keeping the order, and when all products and sums of submatrices involved are defined.Let

Then the matrix product can be performed blockwise, yielding

are calculated by multiplying: Or, using the Einstein notation that implicitly sums over repeated indices: Depicting

as a matrix, we have If a matrix is partitioned into four blocks, it can be inverted blockwise as follows: where A and D are square blocks of arbitrary size, and B and C are conformable with them for partitioning.

[15] Equivalently, by permuting the blocks: Here, D and the Schur complement of D in P: P/D = A − BD−1C must be invertible.

-matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices

are same and equal to the product of characteristic polynomials of

The converse is false; simply check

is invertible, one has If the blocks are square matrices of the same size further formulas hold.

This formula has been generalized to matrices composed of more than

B and defined as For instance, This operation generalizes naturally to arbitrary dimensioned arrays (provided that A and B have the same number of dimensions).

Note that any element in the direct sum of two vector spaces of matrices could be represented as a direct sum of two matrices.

[16] It can also be indicated as A1 ⊕ A2 ⊕ ... ⊕ An[10] or diag(A1, A2, ..., An)[10] (the latter being the same formalism used for a diagonal matrix).

Any square matrix can trivially be considered a block diagonal matrix with only one block.

For the determinant and trace, the following properties hold: A block diagonal matrix is invertible if and only if each of its main-diagonal blocks are invertible, and in this case its inverse is another block diagonal matrix given by The eigenvalues[23] and eigenvectors of

[21] A block tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix a square matrix, having square matrices (blocks) in the lower diagonal, main diagonal and upper diagonal, with all other blocks being zero matrices.

It is essentially a tridiagonal matrix but has submatrices in places of scalars.

are square sub-matrices of the lower, main and upper diagonal respectively.

[24][25] Block tridiagonal matrices are often encountered in numerical solutions of engineering problems (e.g., computational fluid dynamics).

Optimized numerical methods for LU factorization are available[26] and hence efficient solution algorithms for equation systems with a block tridiagonal matrix as coefficient matrix.

The Thomas algorithm, used for efficient solution of equation systems involving a tridiagonal matrix can also be applied using matrix operations to block tridiagonal matrices (see also Block LU decomposition).