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Block matrix

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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]

A 168×168 element block matrix with 12×12, 12×24, 24×12, and 24×24 sub-matrices. Non-zero elements are in blue, zero elements are grayed.


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


Formal definition[edit]

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]

Partitioning methods[edit]

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]

Common partitions[edit]

Often,[9] we encounter the 2x2 partition


particularly in the form where is a scalar:


Block matrix operations[edit]



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]

Block transpose[edit]

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.




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 = DCA−1B must be invertible.[15]

Equivalently, by permuting the blocks:


Here, D and the Schur complement of D in P: P/D = ABD−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 . Furthermore, If or is diagonalizable, then and are diagonalizable too. The converse is false; simply check .

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)


Special types of block matrices[edit]

Direct sums and block diagonal matrices[edit]

Direct sum[edit]

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.

Block diagonal matrices [edit]

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:

[20][21] and

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]

Block tridiagonal matrices[edit]

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).

Block triangular matrices[edit]

Upper block triangular[edit]

A matrix is upper block triangular (or block upper triangular[27]) if


where for all .[23][27]

Lower block triangular[edit]

A matrix is lower block triangular if


where for all .[23]

Block Toeplitz matrices[edit]

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]

Block Hankel matrices[edit]

A matrix is block Hankel if for all , that is,


where .[23]

See also[edit]

  • Kronecker product (matrix direct product resulting in a block matrix)
  • Jordan normal form (canonical form of a linear operator on a finite-dimensional complex vector space)
  • Strassen algorithm (algorithm for matrix multiplication that is faster than the conventional matrix multiplication algorithm)


  1. ^ Eves, Howard (1980). Elementary Matrix Theory (reprint ed.). New York: Dover. p. 37. ISBN 0-486-63946-0. Retrieved 24 April 2013. 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.
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  13. ^ Anton, Howard (1994). Elementary Linear Algebra (7th ed.). New York: John Wiley. p. 36. ISBN 0-471-58742-7. ...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.
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