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.

\[\left[ \begin{array}{ccc|c} a_{11} & a_{12} & a_{13} & b_{1} \\ a_{21} & a_{22} & a_{23} & b_{2} \\ \hline c_{1} & c_{2} & c_{3} & d \end{array} \right]\]

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 \(n\) by \(m\) matrix \(M\) by partitioning \(n\) into a collection \(\text{rowgroups}\), and then partitioning \(m\) into a collection \(\text{colgroups}\). The original matrix is then considered as the "total" of these groups, in the sense that the \((i, j)\) entry of the original matrix corresponds in a 1-to-1 way with some \((s, t)\) offset entry of some \((x,y)\), where \(x \in \text{rowgroups}\) and \(y \in \text{colgroups}\).

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

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. By this partition, \(P\) is partitioned into four 2×2 blocks, as

\[\begin{align} \mathbf{P}_{11} &= \begin{bmatrix} 1 & 2 \\ 1 & 5 \end{bmatrix}, & \mathbf{P}_{12} &= \begin{bmatrix} 2 & 7\\ 6 & 2 \end{bmatrix}, \\[1ex] \mathbf{P}_{21} &= \begin{bmatrix} 3 & 3 \\ 3 & 3 \end{bmatrix},& \mathbf{P}_{22} &= \begin{bmatrix} 4 & 5 \\ 6 & 7 \end{bmatrix}. \end{align}\]

The partitioned matrix can then be written as

Condensed: the full section is in Wikipedia.

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.

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 \(j\)th 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 \(i\)-th row of \(A\).

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:

\[A = \begin{bmatrix} a_{11} & a_{12}^T \\ a_{21} & A_{22} \end{bmatrix}.\]

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 \(A_{ij} \in \mathbb{C}^{k_i \times \ell_j}\). (This matrix \(A\) will be reused in § Addition and § Multiplication.) Then its transpose is

\[A^T = \begin{bmatrix} A_{11}^T & A_{21}^T & \cdots & A_{p1}^T \\ A_{12}^T & A_{22}^T & \cdots & A_{p2}^T \\ \vdots & \vdots & \ddots & \vdots \\ A_{1q}^T & A_{2q}^T & \cdots & A_{pq}^T \end{bmatrix},\]

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

Let

\[B = \begin{bmatrix} B_{11} & B_{12} & \cdots & B_{1s} \\ B_{21} & B_{22} & \cdots & B_{2s} \\ \vdots & \vdots & \ddots & \vdots \\ B_{r1} & B_{r2} & \cdots & B_{rs} \end{bmatrix},\]

where \(B_{ij} \in \mathbb{C}^{m_i \times n_j}\), and let \(A\) be the matrix defined in § Transpose. (This matrix \(B\) will be reused in § Multiplication.) Then if \(p = r\), \(q = s\), \(k_i = m_i\), and \(\ell_j = n_j\), then

\[A + B = \begin{bmatrix} A_{11} + B_{11} & A_{12} + B_{12} & \cdots & A_{1q} + B_{1q} \\ A_{21} + B_{21} & A_{22} + B_{22} & \cdots & A_{2q} + B_{2q} \\ \vdots & \vdots & \ddots & \vdots \\ A_{p1} + B_{p1} & A_{p2} + B_{p2} & \cdots & A_{pq} + B_{pq} \end{bmatrix}.\]

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 \(A\) and \(B\) such that all submatrix products that will be used are defined.

, Arak M. Mathai and Hans J. Haubold, Linear Algebra: A Course for Physicists and Engineers

Let \(A\) be the matrix defined in § Transpose, and let \(B\) be the matrix defined in § Addition. 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:

\[C_{ij} = A_{ik}B_{kj}.\]

Condensed: the full section is in Wikipedia.

If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:

\[\begin{align} P &= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \\[1ex] &= \begin{bmatrix} A^{-1} + A^{-1} B \left(D - CA^{-1} B\right)^{-1} CA^{-1} & -A^{-1} B\left(D - C A^{-1}B\right)^{-1} \\ -\left(D - CA^{-1}B\right)^{-1} C A^{-1} & \left(D - C A^{-1} B\right)^{-1} \end{bmatrix}, \end{align}\]

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 − CAB must be invertible.

Equivalently, by permuting the blocks:

\[\begin{align} P &= \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} \\[1ex] &= \begin{bmatrix} \left(A - BD^{-1}C\right)^{-1} & -\left(A-BD^{-1}C\right)^{-1} BD^{-1} \\ -D^{-1}C\left(A - BD^{-1}C\right)^{-1} & D^{-1} + D^{-1}C\left(A - BD^{-1}C\right)^{-1}BD^{-1} \end{bmatrix}. \end{align}\]

Here, D and the Schur complement of D in P: P/D = A − BDC must be invertible.

If A and D are both invertible, then:

Condensed: the full section is in Wikipedia.

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 \[\det\begin{bmatrix}A& 0\\ C& D\end{bmatrix} = \det(A) \det(D) = \det\begin{bmatrix}A& B\\ 0& D\end{bmatrix}.\]

Using this formula, we can derive that characteristic polynomials of \(\begin{bmatrix}A& 0\\ C& D\end{bmatrix}\) and \(\begin{bmatrix}A& B\\ 0& D\end{bmatrix}\) are same and equal to the product of characteristic polynomials of \(A\) and \(D\). Furthermore, If \(\begin{bmatrix}A& 0\\ C& D\end{bmatrix}\) or \(\begin{bmatrix}A& B\\ 0& D\end{bmatrix}\) is diagonalizable, then \(A\) and \(D\) are diagonalizable too. The converse is false; simply check \(\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}\).

If \(A\) is invertible, one has

\[\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(A) \det\left(D - C A^{-1} B\right),\]

and if \(D\) is invertible, one has

\[\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(D) \det\left(A - B D^{-1} C\right) .\]

If the blocks are square matrices of the same size further formulas hold. For example, if \(C\) and \(D\) commute (i.e., \(CD=DC\)), then \[\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - BC).\] Similar statements hold when \(AB=BA\), \(AC=CA\), or ⁠\(BD=DB\)⁠. 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.

Condensed: the full section is in Wikipedia.

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

\[{A} = \begin{bmatrix} B_1 & C_1 & & & \cdots & & 0 \\ A_2 & B_2 & C_2 & & & & \\ & \ddots & \ddots & \ddots & & & \vdots \\ & & A_k & B_k & C_k & & \\ \vdots & & & \ddots & \ddots & \ddots & \\ & & & & A_{n-1} & B_{n-1} & C_{n-1} \\ 0 & & \cdots & & & A_n & B_n \end{bmatrix}\]

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).

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 \(i, j = 1, \ldots, k\).

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 \(A_i \in \mathbb{F}^{n_i \times m_i}\).

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 \(A_i \in \mathbb{F}^{n_i \times m_i}\).