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. [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. [3] [2] For example, the 3x4 matrix presented below is divided by horizontal and vertical lines into four blocks: the top-left 2x3 block, the top-right 2x1 block, the bottom-left 1x3 block, and the bottom-right 1x1 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.
This notion can be made more precise for an by matrix by partitioning into a collection , and then partitioning into a collection . 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 offset entry of some , where and . [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. [6] [7] By this partition, is partitioned into four 2×2 blocks, as
The partitioned matrix can then be written as
Let . A partitioning of is a representation of in the form
where are contiguous submatrices, , and . [9] The elements of the partition are called blocks. [9]
By this definition, the blocks in any one column must all have the same number of columns. [9] Similarly, the blocks in any one row must have the same number of rows. [9]
A matrix can be partitioned in many ways. [9] For example, a matrix is said to be partitioned by columns if it is written as
where is the th column of . [9] A matrix can also be partitioned by rows:
where is the th row of . [9]
Often, [9] we encounter the 2x2 partition
particularly in the form where is a scalar:
Let
where . (This matrix will be reused in § Addition and § Multiplication.) Then its transpose is
and the same equation holds with the transpose replaced by the conjugate transpose. [9]
A special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed. Let be a block matrix with blocks , the block transpose of is the block matrix with blocks . [11] As with the conventional trace operator, the block transpose is a linear mapping such that . [10] However, in general the property does not hold unless the blocks of and commute.
Let
where , and let be the matrix defined in § Transpose. (This matrix will be reused in § Multiplication.) Then if , , , and , then
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" [12] between two matrices and such that all submatrix products that will be used are defined. [13]
Two matrices and are said to be partitioned conformally for the product , when and are partitioned into submatrices and if the multiplication 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.
— Arak M. Mathai and Hans J. Haubold, Linear Algebra: A Course for Physicists and Engineers [14]
Let be the matrix defined in § Transpose, and let be the matrix defined in § Addition. Then the matrix product
can be performed blockwise, yielding as an matrix. The matrices in the resulting matrix 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. Furthermore, A and the Schur complement of A in P: P/A = D − CA−1B must be invertible. [15]
Equivalently, by permuting the blocks:
Here, D and the Schur complement of D in P: P/D = A − BD−1C 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.
The formula for the determinant of a -matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices . 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 of and are same and equal to the product of characteristic polynomials of and .[ citation needed ] Furthermore, If or is diagonalizable, then and are diagonalizable too. The converse is false; simply check .[ citation needed ]
If is invertible, one has
and if is invertible, one has
If the blocks are square matrices of the same size further formulas hold. For example, if and commute (i.e., ), then
This formula has been generalized to matrices composed of more than blocks, again under appropriate commutativity conditions among the individual blocks. [19]
For and , the following formula holds (even if and do not commute)
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 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.
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. [16] 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. [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 are simply those of the s combined. [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. A block tridiagonal matrix has the form
where , and 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).
A matrix is upper block triangular (or block upper triangular [27] ) if
A matrix is lower block triangular if
where for all . [23]
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.
A matrix is block Toeplitz if for all , that is,
where . [23]
A matrix is block Hankel if for all , that is,
where . [23]
We shall find that it is sometimes convenient to subdivide a matrix into rectangular blocks of elements. This leads us to consider so-called partitioned, or block, matrices.
A matrix can be subdivided or partitioned into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.
A partitioning as in Theorem 1.9.4 is called a conformable partition of A and B.
...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.
In mathematics, the determinant is a scalar value that is a certain function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the linear map represented, on a given basis, by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the corresponding linear map is an isomorphism. The determinant of a product of matrices is the product of their determinants.
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