Block matrix on arvau

Wikipedia / 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. For example, the 3×4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2×3 block, the top-right 2×1 block, the bottom-left 1×3 block, and the bottom-right 1×1 block.

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.

entry of the original matrix corresponds in a 1-to-1 offset.

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

Example

The matrix

can be visualized as divided into four blocks, as

The horizontal and vertical lines have no special mathematical meaning, but are a common way to visualize a partition. is partitioned into four 2×2 blocks, as

The partitioned matrix can then be written as

Formal definition

. A partitioning in the form

. of the partition are called blocks.

By this definition, the blocks in any one column must all have the same number of columns. Similarly, the blocks in any one row must have the same number of rows.

Partitioning methods

A matrix can be partitioned in many ways. is said to be partitioned by columns if it is written as

. A matrix can also be partitioned by rows:

is the .

Common partitions

Often, we encounter the 2×2 partition

is a scalar:

Block matrix operations

Transpose

Let

where will be reused in and .) Then its transpose is

and the same equation holds with the transpose replaced by the conjugate transpose.

Block transpose

A special form of matrix transpose. As with the conventional trace operator, the block transpose is a linear mapping. commute.

Addition

Let

be the matrix defined in will be reused in , then

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" such that all submatrix products that will be used are defined.

be the matrix defined in be the matrix defined in . Then the matrix product

are calculated by multiplying:

Or, using the Einstein notation that implicitly sums over repeated indices:

as a matrix, we have

InversionInversion

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. 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 matrix inversion also enables to yield from the efficiency of the fast matrix multiplication algorithms, Sect. 11, pp. 413-414 .

Computing submatrix inverses from the full inverse

By the symmetry between a matrix and its inverse in the block inversion formula, if a matrix P and its inverse P−1 are partitioned conformally:

then the inverse of any principal submatrix can be computed from the corresponding blocks of P−1:

This relationship follows from recognizing that E−1 = A − BD−1C (the Schur complement), and applying the same block inversion formula with the roles of P and P−1 reversed.

DeterminantDeterminant

square. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is

Using this formula, we can derive that characteristic polynomials is diagonalizable.

is invertible, one has

If the blocks are square matrices of the same commute), then, or . Note for the last two results, you have to use commutativity of the underlying ring, but not for the first two.

blocks, again under appropriate commutativity conditions among the individual blocks.

do not commute)

Special types of block matrices

Direct sums and block diagonal matrices

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 AB 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.

Block diagonal matrices Block diagonal matrix

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 Ak is a square matrix for all k = 1, ..., n. In other words, matrix A is the direct sum of A1, ..., An. 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 eigenvectorss 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 has the form

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 triangular matrices

is upper block triangular (or block upper triangular. is lower block triangular.

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.