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((Short description|Matrix defined using smaller matrices called blocks))
((More citations needed|date=December 2009))
In [[mathematics]], a '''block matrix''' or a '''partitioned matrix''' is a [[matrix (mathematics)|matrix]] that is ''interpreted'' as having been broken into sections called '''blocks''' or '''submatrices'''.<ref>((cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=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''.))</ref> 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 of a set|partition]] it, into a collection of smaller matrices.<ref>((cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=30 |edition=7th |quote=A matrix can be subdivided or '''''partitioned''''' into smaller matrices by inserting horizontal and vertical rules between selected rows and columns.))</ref> 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 <math>n</math> by <math>m</math> matrix <math>M</math> by partitioning <math>n</math> into a collection <math>\text{rowgroups}</math>, and then partitioning <math>m</math> into a collection <math>\text{colgroups}</math>. The original matrix is then considered as the "total" of these groups, in the sense that the <math>(i, j)</math> entry of the original matrix corresponds in a [[Bijection|1-to-1]] way with some <math>(s, t)</math> [[offset (computer science)|offset]] entry of some <math>(x,y)</math>, where <math>x \in \text{rowgroups}</math> and <math>y \in \text{colgroups}</math>.

Block matrix algebra arises in general from [[biproduct]]s in [[Category (mathematics)|categories]] of matrices.<ref>((cite journal | last1 = Macedo | first1 = H.D. | last2 = Oliveira | first2 = J.N. | year = 2013 | title = Typing linear algebra: A biproduct-oriented approach | doi = 10.1016/j.scico.2012.07.012 | journal = Science of Computer Programming | volume = 78 | issue = 11| pages = 2160–2191 | arxiv = 1312.4818 ))</ref>

==Example==
[[File:BlockMatrix168square.png|thumb|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

:<math>\mathbf{P} = \begin{bmatrix}
  1 & 2 & 2 & 7 \\
  1 & 5 & 6 & 2 \\
  3 & 3 & 4 & 5 \\
  3 & 3 & 6 & 7
\end{bmatrix}</math>

can be partitioned into four 2×2 blocks

:<math>
  \mathbf{P}_{11} = \begin{bmatrix}
    1 & 2 \\
    1 & 5
  \end{bmatrix},\quad
  \mathbf{P}_{12} = \begin{bmatrix}
    2 & 7\\
    6 & 2
  \end{bmatrix},\quad
  \mathbf{P}_{21} = \begin{bmatrix}
    3 & 3 \\
    3 & 3
  \end{bmatrix},\quad
  \mathbf{P}_{22} = \begin{bmatrix}
    4 & 5 \\
    6 & 7
  \end{bmatrix}.
</math>

The partitioned matrix can then be written as

:<math>\mathbf{P} = \begin{bmatrix}
  \mathbf{P}_{11} & \mathbf{P}_{12} \\
  \mathbf{P}_{21} & \mathbf{P}_{22}
\end{bmatrix}.</math>

==Block matrix 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 matrix|conformable]] partitions"<ref>((cite book |last=Eves |first=Howard |author-link=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=[https://archive.org/details/elementarymatrix0000eves_r2m2/page/37 37] |url=https://archive.org/details/elementarymatrix0000eves_r2m2 |url-access=registration |edition=reprint |access-date=24 April 2013 |quote=A partitioning as in Theorem 1.9.4 is called a ''conformable partition'' of ''A'' and ''B''.))</ref> between two matrices <math>A</math> and <math>B</math> such that all submatrix products that will be used are defined.<ref>((cite book |last=Anton |first=Howard |title=Elementary Linear Algebra |year=1994 |publisher=John Wiley |location=New York |isbn=0-471-58742-7 |page=36 |edition=7th |quote=...provided the sizes of the submatrices of A and B are such that the indicated operations can be performed.))</ref> Given an <math>(m \times p)</math> matrix <math>\mathbf{A}</math> with <math>q</math> row partitions and <math>s</math> column partitions

:<math>\mathbf{A} = \begin{bmatrix}
  \mathbf{A}_{11} & \mathbf{A}_{12} & \cdots & \mathbf{A}_{1s} \\
  \mathbf{A}_{21} & \mathbf{A}_{22} & \cdots & \mathbf{A}_{2s} \\
  \vdots          & \vdots          & \ddots & \vdots          \\
  \mathbf{A}_{q1} & \mathbf{A}_{q2} & \cdots & \mathbf{A}_{qs}
\end{bmatrix}</math>

and a <math>(p \times n)</math> matrix <math>\mathbf{B}</math> with <math>s</math> row partitions and <math>r</math> column partitions

:<math>\mathbf{B} = \begin{bmatrix}
  \mathbf{B}_{11} & \mathbf{B}_{12} & \cdots &\mathbf{B}_{1r} \\
  \mathbf{B}_{21} & \mathbf{B}_{22} & \cdots &\mathbf{B}_{2r} \\
  \vdots          & \vdots          & \ddots &\vdots          \\
  \mathbf{B}_{s1} & \mathbf{B}_{s2} & \cdots &\mathbf{B}_{sr}
\end{bmatrix},</math>

that are compatible with the partitions of <math>A</math>, the matrix product

:<math>
  \mathbf{C}=\mathbf{A}\mathbf{B}
</math>

can be performed blockwise, yielding <math>\mathbf{C}</math> as an <math>(m \times n)</math> matrix with <math>q</math> row partitions and <math>r</math> column partitions. The matrices in the resulting matrix <math>\mathbf{C}</math> are calculated by multiplying:

:<math>
  \mathbf{C}_{q r} = \sum^s_{i=1}\mathbf{A}_{q i}\mathbf{B}_{i r}. 
</math>

Or, using the [[Einstein notation]] that implicitly sums over repeated indices:

:<math>
  \mathbf{C}_{q r} = \mathbf{A}_{q i}\mathbf{B}_{i r}. 
</math>

==Block matrix inversion((anchor|Inversion))==
((for|more details and derivation using block LDU decomposition|Schur complement))
((see also|Helmert–Wolf blocking))

If a matrix is partitioned into four blocks, it can be [[invertible matrix#Blockwise inversion|inverted blockwise]] as follows:

:<math>\mathbf{P} = \begin{bmatrix}
    \mathbf{A} & \mathbf{B} \\
    \mathbf{C} & \mathbf{D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
     \mathbf{A}^{-1} + \mathbf{A}^{-1}\mathbf{B}\left(\mathbf{D} - \mathbf{CA}^{-1}\mathbf{B}\right)^{-1}\mathbf{CA}^{-1} &
      -\mathbf{A}^{-1}\mathbf{B}\left(\mathbf{D} - \mathbf{CA}^{-1}\mathbf{B}\right)^{-1} \\
    -\left(\mathbf{D}-\mathbf{CA}^{-1}\mathbf{B}\right)^{-1}\mathbf{CA}^{-1} &
       \left(\mathbf{D} - \mathbf{CA}^{-1}\mathbf{B}\right)^{-1}
  \end{bmatrix},
</math>

where '''A''' and '''D''' are square blocks of arbitrary size, and '''B''' and '''C''' are [[conformable matrix|conformable]] with them for partitioning. Furthermore, '''A''' and the Schur complement of '''A''' in '''P''': ((nowrap|'''P'''/'''A''' ((=)) '''D''' − '''CA'''((sup|−1))'''B''')) must be invertible.<ref>
((cite book
  | last = Bernstein
  | first = Dennis
  | title = Matrix Mathematics
  | publisher = Princeton University Press
  | year = 2005
  | pages = 44
  | isbn = 0-691-11802-7
))</ref>

Equivalently, by permuting the blocks:

:<math>\mathbf{P} = \begin{bmatrix}
    \mathbf{A} & \mathbf{B} \\
    \mathbf{C} & \mathbf{D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
     \left(\mathbf{A} - \mathbf{BD}^{-1}\mathbf{C}\right)^{-1} &
      -\left(\mathbf{A}-\mathbf{BD}^{-1}\mathbf{C}\right)^{-1}\mathbf{BD}^{-1} \\
    -\mathbf{D}^{-1}\mathbf{C}\left(\mathbf{A} - \mathbf{BD}^{-1}\mathbf{C}\right)^{-1} &
       \quad \mathbf{D}^{-1} + \mathbf{D}^{-1}\mathbf{C}\left(\mathbf{A} - \mathbf{BD}^{-1}\mathbf{C}\right)^{-1}\mathbf{BD}^{-1}
  \end{bmatrix}.
</math>

Here, '''D''' and the Schur complement of '''D''' in '''P''': ((nowrap|'''P'''/'''D''' ((=)) '''A''' − '''BD'''((sup|−1))'''C''')) must be invertible.

If '''A''' and '''D''' are both invertible, then:

: <math>
  \begin{bmatrix}
    \mathbf{A} & \mathbf{B} \\
    \mathbf{C} & \mathbf{D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
    \left(\mathbf{A} - \mathbf{B} \mathbf{D}^{-1} \mathbf{C}\right)^{-1} & \mathbf{0} \\
    \mathbf{0} & \left(\mathbf{D} - \mathbf{C} \mathbf{A}^{-1} \mathbf{B}\right)^{-1}
  \end{bmatrix} \begin{bmatrix}
                     \mathbf{I} & -\mathbf{B} \mathbf{D}^{-1} \\
    -\mathbf{C} \mathbf{A}^{-1} &                  \mathbf{I}
  \end{bmatrix}.
</math>

By the [[Weinstein–Aronszajn identity]], one of the two matrices in the block-diagonal matrix is invertible exactly when the other is.

==Block matrix determinant((anchor|Determinant))==
The formula for the determinant of a <math>2 \times 2</math>-matrix above continues to hold, under appropriate further assumptions, for a matrix composed of four submatrices <math>A, B, C, D</math>. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the [[Schur complement]], is
:<math>\det\begin{pmatrix}A& 0\\ C& D\end{pmatrix} = \det(A) \det(D) = \det\begin{pmatrix}A& B\\ 0& D\end{pmatrix}.</math>

Using this formula, we can derive that [[characteristic polynomial]]s of <math>\begin{pmatrix}A& 0\\ C& D\end{pmatrix}</math> and <math>\begin{pmatrix}A& B\\ 0& D\end{pmatrix}</math> are same and equal to the product of characteristic polynomials of <math>A</math> and <math>D</math>. Furthermore, If <math>\begin{pmatrix}A& 0\\ C& D\end{pmatrix}</math> or <math>\begin{pmatrix}A& B\\ 0& D\end{pmatrix}</math> is [[diagonalizable]], then <math>A</math> and <math>D</math> are diagonalizable too. The converse is false; simply check <math>\begin{pmatrix}1& 1\\ 0& 1\end{pmatrix}</math>.

If <math>A</math> is [[Invertible matrix|invertible]] (and similarly if <math>D</math> is invertible<ref>Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.</ref>), one has

:<math>\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(A) \det\left(D - C A^{-1} B\right) .</math>
If <math>D</math> is a <math>1 \times 1</math>-matrix, this simplifies to <math>\det (A) (D - CA^{-1}B)</math>.

If the blocks are square matrices of the ''same'' size further formulas hold. For example, if <math>C</math> and <math>D</math> [[commutativity|commute]] (i.e., <math>CD=DC</math>), then 
:<math>\det\begin{pmatrix}A& B\\ C& D\end{pmatrix} = \det(AD - BC).</math><ref>((Cite journal|first= J. R.|last= Silvester|title= Determinants of Block Matrices|journal= Math. Gaz.|volume= 84|issue= 501|year= 2000|pages= 460–467|jstor= 3620776|url= http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf|doi= 10.2307/3620776|access-date= 2021-06-25|archive-date= 2015-03-18|archive-url= https://web.archive.org/web/20150318222335/http://www.ee.iisc.ernet.in/new/people/faculty/prasantg/downloads/blocks.pdf|url-status= dead))</ref>
This formula has been generalized to matrices composed of more than <math>2 \times 2</math> blocks, again under appropriate commutativity conditions among the individual blocks.<ref>((cite journal|last1=Sothanaphan|first1=Nat|title=Determinants of block matrices with noncommuting blocks|journal=Linear Algebra and Its Applications|date=January 2017|volume=512|pages=202–218|doi=10.1016/j.laa.2016.10.004|arxiv=1805.06027|s2cid=119272194))</ref>

For <math>A = D </math> and <math>B=C</math>, the following formula holds (even if <math>A</math> and <math>B</math> do not commute)((citation needed|date=May 2021))
:<math>\det\begin{pmatrix}A& B\\ B& A\end{pmatrix} = \det(A - B) \det(A + B).</math>

==Block diagonal matrices ((anchor|Block diagonal matrix)) ==
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

:<math>\mathbf{A} = \begin{bmatrix} 
  \mathbf{A}_1 & \mathbf{0}    & \cdots & \mathbf{0}   \\
  \mathbf{0}   & \mathbf{A}_2  & \cdots & \mathbf{0}   \\
  \vdots       & \vdots        & \ddots & \vdots       \\
  \mathbf{0}   & \mathbf{0}    & \cdots & \mathbf{A}_n
\end{bmatrix}</math>

where '''A'''<sub>''k''</sub> is a square matrix for all ''k'' = 1, ..., ''n''. In other words, matrix '''A''' is the [[direct sum of matrices|direct sum]] of '''A'''<sub>1</sub>, ..., '''A'''<sub>''n''</sub>. It can also be indicated as '''A'''<sub>1</sub>&nbsp;⊕&nbsp;'''A'''<sub>2</sub>&nbsp;⊕&nbsp;...&nbsp;⊕&nbsp;'''A'''<sub>''n''</sub> or diag('''A'''<sub>1</sub>, '''A'''<sub>2</sub>, ..., '''A'''<sub>''n''</sub>) &nbsp;(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 (linear algebra)|trace]], the following properties hold
:<math>\begin{align}
               \det\mathbf{A} &= \det\mathbf{A}_1 \times \cdots \times \det\mathbf{A}_n, \\
  \operatorname{tr}\mathbf{A} &= \operatorname{tr} \mathbf{A}_1 + \cdots + \operatorname{tr} \mathbf{A}_n.\end{align}</math>

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
:<math>\begin{bmatrix}
    \mathbf{A}_{1} & \mathbf{0}     & \cdots & \mathbf{0} \\
    \mathbf{0}     & \mathbf{A}_{2} & \cdots & \mathbf{0} \\
    \vdots         & \vdots         & \ddots & \vdots \\
    \mathbf{0}     & \mathbf{0}     & \cdots & \mathbf{A}_{n} 
  \end{bmatrix}^{-1} = \begin{bmatrix}
    \mathbf{A}_{1}^{-1} & \mathbf{0}          & \cdots & \mathbf{0} \\
    \mathbf{0}          & \mathbf{A}_{2}^{-1} & \cdots & \mathbf{0} \\
    \vdots              & \vdots              & \ddots & \vdots \\
    \mathbf{0}          & \mathbf{0}          & \cdots & \mathbf{A}_{n}^{-1} 
  \end{bmatrix}.
</math>

The [[eigenvalues and eigenvectors]] of <math>\mathbf{A}</math> are simply those of the <math>\mathbf{A}_k</math>s 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 '''A''' has the form

:<math>\mathbf{A} = \begin{bmatrix}
  \mathbf{B}_{1} & \mathbf{C}_{1} &                &                &           \cdots &                  &      \mathbf{0}  \\
  \mathbf{A}_{2} & \mathbf{B}_{2} & \mathbf{C}_{2} &                &                  &                  &                  \\
                 &         \ddots &         \ddots &         \ddots &                  &                  &           \vdots \\
                 &                & \mathbf{A}_{k} & \mathbf{B}_{k} &   \mathbf{C}_{k} &                  &                  \\
          \vdots &                &                &         \ddots &           \ddots &           \ddots &                  \\
                 &                &                &                & \mathbf{A}_{n-1} & \mathbf{B}_{n-1} & \mathbf{C}_{n-1} \\
      \mathbf{0} &                &         \cdots &                &                  &   \mathbf{A}_{n} &   \mathbf{B}_{n}
\end{bmatrix}</math>

where '''A'''<sub>''k''</sub>, '''B'''<sub>''k''</sub> and '''C'''<sub>''k''</sub> 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 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 block Toeplitz matrix '''A''' has the form

:<math>\mathbf{A} = \begin{bmatrix}
    \mathbf{A}_{(1,1)} &   \mathbf{A}_{(1,2)} &                    &                    &             \cdots & \mathbf{A}_{(1,n-1)} &   \mathbf{A}_{(1,n)} \\
    \mathbf{A}_{(2,1)} &   \mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} &                    &                    &                      & \mathbf{A}_{(1,n-1)} \\
                       &               \ddots &             \ddots &             \ddots &                    &                      &               \vdots \\
                       &                      & \mathbf{A}_{(2,1)} & \mathbf{A}_{(1,1)} & \mathbf{A}_{(1,2)} &                      &                      \\
                \vdots &                      &                    &             \ddots &             \ddots &               \ddots &                      \\
  \mathbf{A}_{(n-1,1)} &                      &                    &                    & \mathbf{A}_{(2,1)} &   \mathbf{A}_{(1,1)} &   \mathbf{A}_{(1,2)} \\
    \mathbf{A}_{(n,1)} & \mathbf{A}_{(n-1,1)} &             \cdots &                    &                    &   \mathbf{A}_{(2,1)} &   \mathbf{A}_{(1,1)}
\end{bmatrix}.</math>

==Block transpose==
A special form of matrix [[transpose]] can also be defined for block matrices, where individual blocks are reordered but not transposed. Let <math>A=(B_{ij})</math> be a <math>k \times l</math> block matrix with <math>m \times n</math> blocks <math>B_{ij}</math>, the block transpose of <math>A</math> is the <math>l \times k</math> block matrix <math>A^\mathcal{B}</math> with <math>m \times n</math> blocks <math>\left(A^\mathcal{B}\right)_{ij} = B_{ji}</math>.<ref>((cite thesis |last=Mackey |first=D. Steven |date=2006 |title=Structured linearizations for matrix polynomials |publisher=University of Manchester |issn=1749-9097 |oclc=930686781 |url=http://eprints.maths.manchester.ac.uk/314/1/mackey06.pdf))</ref>

As with the conventional trace operator, the block transpose is a [[linear mapping]] such that <math>(A + C)^\mathcal{B} = A^\mathcal{B} + C^\mathcal{B} </math>. However, in general the property <math>(A C)^\mathcal{B} = C^\mathcal{B} A^\mathcal{B} </math> does not hold unless the blocks of <math>A</math> and <math>C</math> commute.

==Direct sum==
((See also|Matrix addition#Direct sum))
For any arbitrary matrices '''A''' (of size ''m''&nbsp;×&nbsp;''n'') and '''B''' (of size ''p''&nbsp;×&nbsp;''q''), we have the '''direct sum''' of '''A''' and '''B''', denoted by '''A'''&nbsp;<math>\oplus</math>&nbsp;'''B''' and defined as 
 
:<math>
  \mathbf{A} \oplus \mathbf{B} =
  \begin{bmatrix}
    a_{11} & \cdots & a_{1n} &      0 & \cdots &      0 \\
    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
    a_{m1} & \cdots & a_{mn} &      0 & \cdots &      0 \\
         0 & \cdots &      0 & b_{11} & \cdots & b_{1q} \\
    \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\
         0 & \cdots &      0 & b_{p1} & \cdots & b_{pq} 
  \end{bmatrix}.
</math>

For instance,

:<math>
  \begin{bmatrix}
    1 & 3 & 2 \\
    2 & 3 & 1
  \end{bmatrix} \oplus
  \begin{bmatrix}
    1 & 6 \\
    0 & 1
  \end{bmatrix} =
  \begin{bmatrix}
    1 & 3 & 2 & 0 & 0 \\
    2 & 3 & 1 & 0 & 0 \\
    0 & 0 & 0 & 1 & 6 \\
    0 & 0 & 0 & 0 & 1
  \end{bmatrix}.
</math>

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 vector spaces|direct sum]] of two [[vector space]]s of matrices could be represented as a direct sum of two matrices.

==Application==
In [[linear algebra]] terms, the use of a block matrix corresponds to having a [[linear mapping]] thought of in terms of corresponding 'bunches' of [[basis vector]]s. That again matches the idea of having distinguished direct sum decompositions of the [[domain of a function|domain]] and [[range of a function|range]]. It is always particularly significant if a block is the [[zero matrix]]; that carries the information that a summand maps into a sub-sum.

Given the interpretation ''via'' linear mappings and direct sums, there is a special type of block matrix that occurs for square matrices (the case ''m'' = ''n''). For those we can assume an interpretation as an [[endomorphism]] of an ''n''-dimensional space ''V''; the block structure in which the bunching of rows and columns is the same is of importance because it corresponds to having a single direct sum decomposition on ''V'' (rather than two). In that case, for example, the [[diagonal]] blocks in the obvious sense are all square. This type of structure is required to describe the [[Jordan normal form]].

This technique is used to cut down calculations of matrices, column-row expansions, and many [[computer science]] applications, including [[VLSI]] chip design. An example is the [[Strassen algorithm]] for fast [[matrix multiplication]], as well as the [[Hamming(7,4)]] encoding for error detection and recovery in data transmissions.

The technique can also be used where the elements of the A,B,C, and D matrices do not all require the same field for their elements.  For example, the matrix A can be over the field of complex numbers, while the matrix D can be over the field of real numbers.   This can lead to valid operations involving the matrices, while simplifying the operations within one of the matrices.  For example,  if D has only real elements finding its inverse takes less calculations than if complex elements must be considered.  But the reals is a subfield of the complex numbers (further it can be considered a projection),  so the matrices operations can be well defined.

==See also==
*[[Kronecker product]] (matrix direct product resulting in a block matrix)

==Notes==
((Reflist))

==References==
*((Cite web |last=Strang |first=Gilbert |author-link=Gilbert Strang |url=http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-3-multiplication-and-inverse-matrices |title=Lecture 3: Multiplication and inverse matrices |publisher=MIT Open Course ware |at=18:30–21:10 |date=1999))

((Linear algebra))
((Matrix classes))

[[Category:Matrices]]
[[Category:Sparse matrices]]

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