Block matrix

In mathematics, a block matrix or a partitioned matrix is a matrix that is "interpreted" as having been broken into sections called blocks or submatrices. 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.
This notion can be made more precise for an formula_1 by formula_2 matrix formula_3 by partitioning formula_1 into a collection formula_5, and then partitioning formula_2 into a collection formula_7. The original matrix is then considered as the "total" of these groups, in the sense that the formula_8 entry of the original matrix corresponds in a 1-to-1 way with some formula_9 offset entry of some formula_10, where formula_11 and formula_12.
Block matrix algebra arises in general from biproducts in categories of matrices.
Example.
The matrix
can be partitioned into four 2×2 blocks
The partitioned matrix can then be written as
Block matrix multiplication.
It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions" between two matrices formula_16 and formula_17 such that all submatrix products that will be used are defined. Given an formula_18 matrix formula_19 with formula_20 row partitions and formula_21 column partitions
and a formula_23 matrix formula_24 with formula_21 row partitions and formula_26 column partitions
that are compatible with the partitions of formula_16, the matrix product
can be formed blockwise, yielding formula_30 as an formula_31 matrix with formula_20 row partitions and formula_26 column partitions. The matrices in the resulting matrix formula_30 are calculated by multiplying:
Or, using the Einstein notation that implicitly sums over repeated indices:
Block matrix inversion.
If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:
where A and D are square of arbitrary size, and B and C are conformable for partitioning. Furthermore, A and the Schur complement of A in P: must be invertible.
Equivalently, by permuting the blocks:
Here, D and the Schur complement of D in P: must be invertible. 
If A and D are both invertible, then:
By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
Block diagonal matrices.
A block diagonal matrix is a block matrix that is a square matrix such that the main-diagonal blocks are square matrices and all off-diagonal blocks are zero matrices. That is, a block diagonal matrix A has the form
where A"k" is a square matrix for all "k" = 1, ..., "n". In other words, matrix A is the direct sum of A1, ..., A"n". It can also be indicated as A1 ⊕ A2 ⊕ ... ⊕ An  or  diag(A1, A2, ..., An)  (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 and eigenvectors of formula_16 are simply those of formula_44 and formula_45 and ... and formula_46 combined.
Block tridiagonal matrices.
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. A block tridiagonal matrix A has the form
where A"k", B"k" and C"k" are square sub-matrices of the lower, main and upper diagonal respectively.
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 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).
Block Toeplitz matrices.
A block Toeplitz matrix is another special block matrix, which contains blocks that are repeated down the diagonals of the matrix, as a Toeplitz matrix has elements repeated down the diagonal. The individual block matrix elements, Aij, must also be a Toeplitz matrix.
A block Toeplitz matrix A has the form
Block transpose.
A special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed. Let formula_49 be a formula_50 block matrix with formula_51 blocks formula_52, the block transpose of formula_16 is the formula_54 block matrix formula_55 with formula_51 blocks formula_57.
As with the conventional trace operator, the block transpose is a linear mapping such that formula_58. However in general the property formula_59 does not hold unless the blocks of formula_16 and formula_61 commute.
Direct sum.
For any arbitrary matrices A (of size "m" × "n") and B (of size "p" × "q"), we have the direct sum of A and B, denoted by A formula_62 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.
Application.
In linear algebra terms, the use of a block matrix corresponds to having a linear mapping thought of in terms of corresponding 'bunches' of basis vectors. That again matches the idea of having distinguished direct sum decompositions of the domain and range. It is always particularly significant if a block is the zero matrix; that carries the information that a summand maps into a sub-sum.
Given the interpretation "via" linear mappings and direct sums, there is a special type of block matrix that occurs for square matrices (the case "m" = "n"). For those we can assume an interpretation as an endomorphism of an "n"-dimensional space "V"; the block structure in which the bunching of rows and columns is the same is of importance because it corresponds to having a single direct sum decomposition on "V" (rather than two). In that case, for example, the diagonal blocks in the obvious sense are all square. This type of structure is required to describe the Jordan normal form.
This technique is used to cut down calculations of matrices, column-row expansions, and many computer science applications, including VLSI chip design. An example is the Strassen algorithm for fast matrix multiplication, as well as the Hamming(7,4) encoding for error detection and recovery in data transmissions.