Matrix defined using smaller matrices called blocks
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 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.
formula_1
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 formula_2 by formula_3 matrix formula_4 by partitioning formula_2 into a collection formula_6, and then partitioning formula_3 into a collection formula_8. The original matrix is then considered as the "total" of these groups, in the sense that the formula_9 entry of the original matrix corresponds in a 1-to-1 way with some formula_10 offset entry of some formula_11, where formula_12 and formula_13.
Block matrix algebra arises in general from biproducts in categories of matrices.
Example.
The matrix
formula_14
can be visualized as divided into four blocks, as 
formula_15.
The horizontal and vertical lines have no special mathematical meaning, but are a common way to visualize a partition. By this partition, formula_16 is partitioned into four 2×2 blocks, as
formula_17
The partitioned matrix can then be written as
formula_18
Formal definition.
Let formula_19. A partitioning of formula_20 is a representation of formula_20 in the form
formula_22,
where formula_23 are contiguous submatrices, formula_24, and formula_25. The elements formula_26 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 formula_20 is said to be partitioned by columns if it is written as
formula_28,
where formula_29 is the formula_30th column of formula_20. A matrix can also be partitioned by rows:
formula_32,
where formula_33 is the formula_34th row of formula_20.
Common partitions.
Often, we encounter the 2x2 partition
formula_36,
particularly in the form where formula_37 is a scalar:
formula_38.
Block matrix operations.
Transpose.
Let
formula_22
where formula_40. (This matrix formula_20 will be reused in and .) Then its transpose is
formula_42,
and the same equation holds with the transpose replaced by the conjugate transpose.
Block transpose.
A special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed. Let formula_43 be a formula_44 block matrix with formula_45 blocks formula_46, the block transpose of formula_20 is the formula_48 block matrix formula_49 with formula_45 blocks formula_51. As with the conventional trace operator, the block transpose is a linear mapping such that formula_52. However, in general the property formula_53 does not hold unless the blocks of formula_20 and formula_55 commute.
Addition.
Let 
formula_56,
where formula_57, and let formula_20 be the matrix defined in . (This matrix formula_59 will be reused in .) Then if formula_60, formula_61, formula_62, and formula_63, then
formula_64.
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" between two matrices formula_20 and formula_59 such that all submatrix products that will be used are defined.
Let formula_20 be the matrix defined in , and let formula_59 be the matrix defined in . Then the matrix product
formula_69
can be performed blockwise, yielding formula_55 as an formula_71 matrix. The matrices in the resulting matrix formula_55 are calculated by multiplying:
formula_73
Or, using the Einstein notation that implicitly sums over repeated indices:
formula_74
Depicting formula_55 as a matrix, we have
formula_76.
Inversion.
If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:
formula_77
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.
Equivalently, by permuting the blocks:
formula_78
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:
 formula_79
By the Weinstein–Aronszajn identity, one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.
Determinant.
The formula for the determinant of a formula_80-matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices formula_81. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is
formula_82
Using this formula, we can derive that characteristic polynomials of formula_83 and formula_84 are same and equal to the product of characteristic polynomials of formula_20 and formula_86. Furthermore, If formula_83 or formula_84 is diagonalizable, then formula_20 and formula_86 are diagonalizable too. The converse is false; simply check formula_91.
If formula_20 is invertible, one has
formula_93
and if formula_86 is invertible, one has
formula_95
If the blocks are square matrices of the "same" size further formulas hold. For example, if formula_55 and formula_86 commute (i.e., formula_98), then 
formula_99
This is also true when formula_100, formula_101, or &amp;NoBreak;&amp;NoBreak;.
This formula has been generalized to matrices composed of more than formula_80 blocks, again under appropriate commutativity conditions among the individual blocks.
For formula_103 and formula_104, the following formula holds (even if formula_20 and formula_59 do not commute)
formula_107
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 formula_108 B and defined as 
formula_109
For instance,
formula_110
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.
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
formula_111
where A"k" is a square matrix for all "k" = 1, ..., "n". In other words, matrix A is the direct sum of A1, ..., A"n". It can also be indicated as A1 ⊕ A2 ⊕ ... ⊕ A"n" or diag(A1, A2, ..., A"n") (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:
formula_112 and
formula_113
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
formula_114
The eigenvalues and eigenvectors of formula_115 are simply those of the formula_116s 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 but has submatrices in places of scalars. A block tridiagonal matrix formula_20 has the form
formula_118
where formula_119, formula_120 and formula_121 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.
Upper block triangular.
A matrix formula_20 is upper block triangular (or block upper triangular) if
formula_123,
where formula_124 for all formula_125.
Lower block triangular.
A matrix formula_20 is lower block triangular if
formula_127,
where formula_124 for all formula_125.
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 formula_20 is block Toeplitz if formula_131 for all formula_132, that is,
formula_133,
where formula_134.
Block Hankel matrices.
A matrix formula_20 is block Hankel if formula_131 for all formula_137, that is,
formula_138,
where formula_134.
Notes.
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