Block matrix

The matrix \mathbf{P} = \begin{bmatrix} 1 & 2 & 2 & 7 \\ 1 & 5 & 6 & 2 \\ 3 & 3 & 4 & 5 \\ 3 & 3 & 6 & 7 \end{bmatrix} can be visualized as divided into four blocks, as \mathbf{P} = \left[ \begin{array}{cc|cc} 1 & 2 & 2 & 7 \\ 1 & 5 & 6 & 2 \\ \hline 3 & 3 & 4 & 5 \\ 3 & 3 & 6 & 7 \end{array} \right]. The horizontal and vertical lines have no special mathematical meaning, but are a common way to visualize a partition. \mathbf{P} = \begin{bmatrix} \mathbf{P}_{11} & \mathbf{P}_{12} \\ \mathbf{P}_{21} & \mathbf{P}_{22} \end{bmatrix}. ==Formal definition==

Let A \in \mathbb{C}^{m \times n}. A partitioning of A is a representation of A in the form A = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1q} \\ A_{21} & A_{22} & \cdots & A_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ A_{p1} & A_{p2} & \cdots & A_{pq} \end{bmatrix}, where A_{ij} \in \mathbb{C}^{m_i \times n_j} are contiguous submatrices, \sum_{i=1}^{p} m_i = m, and \sum_{j=1}^{q} n_j = n. The elements A_{ij} 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. For example, a matrix A is said to be partitioned by columns if it is written as A = (a_1 \ a_2 \ \cdots \ a_n), where a_j is the jth column of A. A matrix can also be partitioned by rows: A = \begin{bmatrix} a_1^T \\ a_2^T \\ \vdots \\ a_m^T \end{bmatrix}, where a_i^T is the row of A. Common partitions Often, we encounter the 2×2 partition A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix}, particularly in the form where A_{11} is a scalar: A = \begin{bmatrix} a_{11} & a_{12}^T \\ a_{21} & A_{22} \end{bmatrix}. ==Block matrix operations==

Transpose Let A = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1q} \\ A_{21} & A_{22} & \cdots & A_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ A_{p1} & A_{p2} & \cdots & A_{pq} \end{bmatrix} where {{nowrap|A_{ij} \in \mathbb{C}^{k_i \times \ell_j}.}} (This matrix A will be reused in and .) Then its transpose is As with the conventional trace operator, the block transpose is a linear mapping such that (A + C)^\mathcal{B} = A^\mathcal{B} + C^\mathcal{B} . between two matrices A and B such that all submatrix products that will be used are defined. Let A be the matrix defined in , and let B be the matrix defined in . Then the matrix product C = AB can be performed blockwise, yielding C as an (p \times s) matrix. The matrices in the resulting matrix C are calculated by multiplying: C_{ij} = \sum_{k=1}^{q} A_{ik}B_{kj}. Or, using the Einstein notation that implicitly sums over repeated indices: C_{ij} = A_{ik}B_{kj}. Depicting C as a matrix, we have Equivalently, by permuting the blocks:, 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: P = \begin{bmatrix} {A} & {B} \\ {C} & {D} \end{bmatrix}, \quad P^{-1} = \begin{bmatrix} {E} & {F} \\ {G} & {H} \end{bmatrix} then the inverse of any principal submatrix can be computed from the corresponding blocks of P−1: {A}^{-1} = {E} - {FH}^{-1}{G} {D}^{-1} = {H} - {GE}^{-1}{F} 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. Determinant The formula for the determinant of a 2 \times 2-matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices A, B, C, D with A and D square. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is \det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - BC). Similar statements hold when AB=BA, AC=CA, or . Namely, if AC = CA, then \det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - CB). Note the change in order of C and B (we have CB instead of BC). Similarly, if BD = DB, then AD should be replaced with DA (i.e. we get \det(DA - BC)) and if AB = BA, then we should have \det(DA - CB). Note for the last two results, you have to use commutativity of the underlying ring, but not for the first two. This formula has been generalized to matrices composed of more than 2 \times 2 blocks, again under appropriate commutativity conditions among the individual blocks. For A = D and B=C, the following formula holds (even if A and B do not commute) \det\begin{bmatrix}A& B\\ B& A\end{bmatrix} = \det(A - B) \det(A + B). ==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 A ⊕ B and defined as That is, a block diagonal matrix A has the form {A} = \begin{bmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_n \end{bmatrix} where Ak is a square matrix for all k = 1, ..., n. In other words, matrix A is the direct sum of A1, ..., 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: :\begin{align} \det{A} &= \det{A}_1 \times \cdots \times \det{A}_n, \end{align} and :\begin{align} \operatorname{tr}{A} &= \operatorname{tr} {A}_1 + \cdots + \operatorname{tr} {A}_n.\end{align} \begin{bmatrix} {A}_{1} & {0} & \cdots & {0} \\ {0} & {A}_{2} & \cdots & {0} \\ \vdots & \vdots & \ddots & \vdots \\ {0} & {0} & \cdots & {A}_{n} \end{bmatrix}^{-1} = \begin{bmatrix} {A}_{1}^{-1} & {0} & \cdots & {0} \\ {0} & {A}_{2}^{-1} & \cdots & {0} \\ \vdots & \vdots & \ddots & \vdots \\ {0} & {0} & \cdots & {A}_{n}^{-1} \end{bmatrix}. The eigenvalues 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 An n \times n matrix A is upper block triangular (or block upper triangular) if there are positive integers n_1, \ldots, n_k such that n = n_1 + n_2 + \ldots + n_k and A = \begin{bmatrix} A_{11} & A_{12} & \cdots & A_{1k} \\ 0 & A_{22} & \cdots & A_{2k} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_{kk} \end{bmatrix}, where the matrix A_{ij} is n_i \times n_j for all i, j = 1, \ldots, k. Similarly, A is lower block triangular if A = \begin{bmatrix} A_{11} & 0 & \cdots & 0 \\ A_{21} & A_{22} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ A_{k1} & A_{k2} & \cdots & A_{kk} \end{bmatrix}, where A_{ij} is n_i \times n_j for all i, j = 1, \ldots, k. 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. A matrix A is block Toeplitz if A_{(i,j)} = A_{(k,l)} for all k - i = l - j, that is, A = \begin{bmatrix} A_1 & A_2 & A_3 & \cdots \\ A_4 & A_1 & A_2 & \cdots \\ A_5 & A_4 & A_1 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}, where A_i \in \mathbb{F}^{n_i \times m_i}. Block Hankel matrices A matrix A is block Hankel if A_{(i,j)} = A_{(k,l)} for all i + j = k + l, that is, A = \begin{bmatrix} A_1 & A_2 & A_3 & \cdots \\ A_2 & A_3 & A_4 & \cdots \\ A_3 & A_4 & A_5 & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix}, where A_i \in \mathbb{F}^{n_i \times m_i}. ==See also==