Block matrix

{{Short description|Matrix defined using smaller matrices called blocks}} 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><ref name=":8">{{Cite web |last=Dobrushkin |first=Vladimir |date= |title=Partition Matrices |url=https://www.cfm.brown.edu/people/dobrush/cs52/Mathematica/Part2/partition.html |access-date=2024-03-24 |website=Linear Algebra with Mathematica}}</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><ref name=":8" /> 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.

<math display="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] </math>

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>.<ref>{{Cite journal |last1=Indhumathi |first1=D. |last2=Sarala |first2=S. |date=2014-05-16 |title=Fragment Analysis and Test Case Generation using F-Measure for Adaptive Random Testing and Partitioned Block based Adaptive Random Testing |url=http://research.ijcaonline.org/volume93/number6/pxc3895662.pdf |journal=International Journal of Computer Applications |volume=93 |issue=6 |pages=13 |doi=10.5120/16218-5662|bibcode=2014IJCA...93f..11I }}</ref>

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> [[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.]]

==Example== The matrix

<math display="block">\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 visualized as divided into four blocks, as

<math display="block">\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].</math>

The horizontal and vertical lines have no special mathematical meaning,<ref name=":3" /><ref name=":4">{{Cite book |last=Johnston |first=Nathaniel |title=Advanced linear and matrix algebra |date=2021 |publisher=Springer Nature |isbn=978-3-030-52814-0 |location=Cham, Switzerland |pages=298}}</ref> but are a common way to visualize a partition.<ref name=":3" /><ref name=":4" /> By this partition, <math>P</math> is partitioned into four 2×2 blocks, as

<math display="block"> \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} </math>

The partitioned matrix can then be written as<ref>{{Cite book |last=Jeffrey |first=Alan |url=https://www.worldcat.org/title/639165077 |title=Matrix operations for engineers and scientists: an essential guide in linear algebra |date=2010 |publisher=Springer |isbn=978-90-481-9273-1 |location=Dordrecht [Netherlands]; New York |pages=54 |oclc=639165077}}</ref>

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

==Formal definition== Let <math>A \in \mathbb{C}^{m \times n}</math>. A '''partitioning''' of <math>A</math> is a representation of <math>A</math> in the form

<math display="block">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},</math>

where <math>A_{ij} \in \mathbb{C}^{m_i \times n_j}</math> are contiguous submatrices, <math display="inline">\sum_{i=1}^{p} m_i = m</math>, and <math display="inline">\sum_{j=1}^{q} n_j = n</math>.<ref name=":2">{{Cite book |last=Stewart |first=Gilbert W. |title=Matrix algorithms. 1: Basic decompositions |date=1998 |publisher=Soc. for Industrial and Applied Mathematics |isbn=978-0-89871-414-2 |location=Philadelphia, PA |pages=18–20}}</ref> The elements <math>A_{ij}</math> of the partition are called '''blocks'''.<ref name=":2" />

By this definition, the blocks in any one column must all have the same number of columns.<ref name=":2" /> Similarly, the blocks in any one row must have the same number of rows.<ref name=":2" />

=== Partitioning methods === A matrix can be partitioned in many ways.<ref name=":2" /> For example, a matrix <math>A</math> is said to be '''partitioned by columns''' if it is written as

<math display="block">A = (a_1 \ a_2 \ \cdots \ a_n),</math>

where <math>a_j</math> is the <math>j</math>th column of <math>A</math>.<ref name=":2" /> A matrix can also be '''partitioned by rows''':

<math display="block">A = \begin{bmatrix} a_1^T \\ a_2^T \\ \vdots \\ a_m^T \end{bmatrix}, </math>

where <math>a_i^T</math> is the {{nowrap|<math>i</math>-th}} row of <math>A</math>.<ref name=":2" />

=== Common partitions === Often,<ref name=":2" /> we encounter the 2×2 partition

<math display="block">A = \begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix},</math>

particularly in the form where <math>A_{11}</math> is a scalar:<ref name=":2" />

<math display="block">A = \begin{bmatrix} a_{11} & a_{12}^T \\ a_{21} & A_{22} \end{bmatrix}.</math>

==Block matrix operations==

===Transpose=== Let

where {{nowrap|<math>A_{ij} \in \mathbb{C}^{k_i \times \ell_j}</math>.}} (This matrix <math>A</math> will be reused in {{section link||Addition}} and {{section link||Multiplication}}.) Then its transpose is<ref name=":2" /><ref name=":1" />

<math display="block">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},</math>

and the same equation holds with the transpose replaced by the conjugate transpose.<ref name=":2" />

====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>.<ref name=":1" /> 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.

===Addition===

Let

<math display="block">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},</math>

where <math>B_{ij} \in \mathbb{C}^{m_i \times n_j}</math>, and let <math>A</math> be the matrix defined in {{section link||Transpose}}. (This matrix <math>B</math> will be reused in {{section link||Multiplication}}.) Then if <math>p = r</math>, <math>q = s</math>, <math>k_i = m_i</math>, and <math>\ell_j = n_j</math>, then<ref name=":2" />

<math display="block">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}.</math>

===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> {{Cquote | quote = Two matrices <math>A</math> and <math>B</math> are said to be partitioned conformally for the product <math>AB</math>, when <math>A</math> and <math>B</math> are partitioned into submatrices and if the multiplication <math>AB</math> 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. | author = Arak M. Mathai and Hans J. Haubold | source = ''Linear Algebra: A Course for Physicists and Engineers''<ref>{{Cite book |last1=Mathai |first1=Arakaparampil M. |title=Linear Algebra: a course for physicists and engineers |last2=Haubold |first2=Hans J. |date=2017 |publisher=De Gruyter |isbn=978-3-11-056259-0 |series=De Gruyter textbook |location=Berlin Boston |pages=162}}</ref> }}

Let <math>A</math> be the matrix defined in {{section link||Transpose}}, and let <math>B</math> be the matrix defined in {{section link||Addition}}. Then the matrix product

can be performed blockwise, yielding <math>C</math> as an <math>(p \times s)</math> matrix. The matrices in the resulting matrix <math>C</math> are calculated by multiplying:<ref name=":3">{{Cite book |last=Johnston |first=Nathaniel |title=Introduction to linear and matrix algebra |date=2021 |publisher=Springer Nature |isbn=978-3-030-52811-9 |location=Cham, Switzerland |pages=30,425}}</ref>

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

Depicting <math>C</math> as a matrix, we have<ref name=":2" />

<math display="block">C = AB = \begin{bmatrix} \sum_{i=1}^{q} A_{1i}B_{i1} & \sum_{i=1}^{q} A_{1i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{1i}B_{is} \\ \sum_{i=1}^{q} A_{2i}B_{i1} & \sum_{i=1}^{q} A_{2i}B_{i2} & \cdots & \sum_{i=1}^{q} A_{2i}B_{is} \\ \vdots & \vdots & \ddots & \vdots \\ \sum_{i=1}^{q} A_{pi}B_{i1} & \sum_{i=1}^{q} A_{pi}B_{i2} & \cdots & \sum_{i=1}^{q} A_{pi}B_{is} \end{bmatrix}.</math>

===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 display="block">\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}</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''': {{math|1='''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:<ref name=":0" />

<math display="block">\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} </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 display="block"> \begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} \left(A - B D^{-1} C\right)^{-1} & 0 \\ 0 & \left(D - C A^{-1} B\right)^{-1} \end{bmatrix} \begin{bmatrix} I & -B D^{-1} \\ -C A^{-1} & 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 inversion also enables to yield from the efficiency of the [[Computational complexity of matrix multiplication|fast matrix multiplication algorithms]], which allows to perform the inversion in time <math>O({n^\omega })</math> for <math>~2.37 \le \omega < 3</math><ref>{{cite journal |last1=Pan |first1=V. |date=1984 |title=How can we speed up matrix multiplication?|journal= SIAM Review |volume=26 |issue=3 |pages=393–415 | doi=10.1137/1026076}}</ref>, 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:

<math display="block">P = \begin{bmatrix} {A} & {B} \\ {C} & {D} \end{bmatrix}, \quad P^{-1} = \begin{bmatrix} {E} & {F} \\ {G} & {H} \end{bmatrix}</math>

then the inverse of any principal submatrix can be computed from the corresponding blocks of '''P'''−1:

This relationship follows from recognizing that '''E'''−1 = '''A''' − '''BD'''−1'''C''' (the Schur complement), and applying the same block inversion formula with the roles of '''P''' and '''P'''−1 reversed.<ref>{{cite web|title=Is this formula for a matrix block inverse in terms of the entire matrix inverse known?|url=https://mathoverflow.net/questions/495299/is-this-formula-for-a-matrix-block-inverse-in-terms-of-the-entire-matrix-inverse|website=MathOverflow}}</ref><ref>{{cite journal|last1=Escalante-B.|first1=Alberto N.|last2=Wiskott|first2=Laurenz|title=Improved graph-based SFA: Information preservation complements the slowness principle|journal=Machine Learning|year=2016|volume=|issue=|pages=|doi=10.1007/s10994-016-5563-y|url=https://www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S1405-55462016000200251|arxiv=1412.4679}}</ref>

===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> with <math>A</math> and <math>D</math> square. The easiest such formula, which can be proven using either the [[Leibniz formula for determinants|Leibniz formula]] or a factorization involving the [[Schur complement]], is<ref name=":0" /> <math display="block">\det\begin{bmatrix}A& 0\\ C& D\end{bmatrix} = \det(A) \det(D) = \det\begin{bmatrix}A& B\\ 0& D\end{bmatrix}.</math>

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

If <math>A</math> is [[Invertible matrix|invertible]], one has<ref name=":0" />

<math display="block">\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(A) \det\left(D - C A^{-1} B\right),</math>

and if <math>D</math> is invertible, one has<ref>Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.</ref><ref name=":0" />

<math display="block">\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(D) \det\left(A - B D^{-1} C\right) .</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<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> <math display="block">\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - BC).</math> Similar statements hold when <math>AB=BA</math>, <math>AC=CA</math>, or {{tmath|1=BD=DB}}. Namely, if <math>AC = CA</math>, then <math display="block">\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(AD - CB).</math> Note the change in order of <math>C</math> and <math>B</math> (we have <math>CB</math> instead of <math>BC</math>). Similarly, if <math>BD = DB</math>, then <math>AD</math> should be replaced with <math>DA</math> (i.e. we get <math>\det(DA - BC)</math>) and if <math>AB = BA</math>, then we should have <math>\det(DA - CB)</math>. 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 <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)<ref name=":0" /> <math display="block">\det\begin{bmatrix}A& B\\ B& A\end{bmatrix} = \det(A - B) \det(A + B).</math>

==Special types of block matrices==

===Direct sums and block diagonal matrices===

====Direct sum==== {{See also|Direct sum of matrices}}

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''' &oplus; '''B''' and defined as<ref name=":1" />

<math display="block"> {A} \oplus {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 display="block"> \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.

====Block diagonal matrices {{anchor|Block diagonal matrix}} ==== {{See also|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.<ref name=":0">{{Cite book |last1=Abadir |first1=Karim M. |title=Matrix Algebra |last2=Magnus |first2=Jan R. |publisher=Cambridge University Press |year=2005 |isbn=9781139443647 |pages=97,100,106,111,114,118 |language=en}}</ref> That is, a block diagonal matrix '''A''' has the form

<math display="block">{A} = \begin{bmatrix} A_1 & 0 & \cdots & 0 \\ 0 & A_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & A_n \end{bmatrix}</math>

where '''A'''''k'' is a square matrix for all ''k'' = 1, ..., ''n''. In other words, matrix '''A''' is the [[direct sum of matrices|direct sum]] of '''A'''1, ..., '''A'''''n''.<ref name=":0" /> It can also be indicated as '''A'''1 ⊕ '''A'''2 ⊕ ... ⊕ '''A'''''n''<ref name=":1" /> or diag('''A'''1, '''A'''2, ..., '''A'''''n'')<ref name=":1">{{Cite book |last=Gentle |first=James E. |title=Matrix Algebra: Theory, Computations, and Applications in Statistics |date=2007 |publisher=Springer New York Springer e-books |isbn=978-0-387-70873-7 |series=Springer Texts in Statistics |location=New York, NY |pages=47,487}}</ref> (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{A} &= \det{A}_1 \times \cdots \times \det{A}_n, \end{align}</math><ref>{{Cite book |last1=Quarteroni |first1=Alfio |title=Numerical mathematics |last2=Sacco |first2=Riccardo |last3=Saleri |first3=Fausto |date=2000 |publisher=Springer |isbn=978-0-387-98959-4 |series=Texts in applied mathematics |location=New York |pages=10, 13}}</ref><ref name=":6">{{Cite journal |last1=George |first1=Raju K. |last2=Ajayakumar |first2=Abhijith |date=2024 |title=A Course in Linear Algebra |url=https://doi.org/10.1007/978-981-99-8680-4 |journal=University Texts in the Mathematical Sciences |language=en |pages=35,407 |doi=10.1007/978-981-99-8680-4 |isbn=978-981-99-8679-8 |issn=2731-9318|url-access=subscription }}</ref> and :<math>\begin{align} \operatorname{tr}{A} &= \operatorname{tr} {A}_1 + \cdots + \operatorname{tr} {A}_n.\end{align}</math><ref name=":0" /><ref name=":6" />

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<ref>{{Cite book |last=Prince |first=Simon J. D. |title=Computer vision: models, learning, and inference |date=2012 |publisher=Cambridge university press |isbn=978-1-107-01179-3 |location=New York |pages=531}}</ref> <math display="block">\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}. </math>

The [[eigenvalues and eigenvectors|eigenvalues]]<ref name=":5" /> [[eigenvalues and eigenvectors|and eigenvectors]] of <math>{A}</math> are simply those of the <math>{A}_k</math>s combined.<ref name=":6" />

===Block tridiagonal matrices=== {{See also|Tridiagonal matrix}} 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 <math>A</math> has the form

<math display="block">{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}</math>

where <math>{A}_{k}</math>, <math>{B}_{k}</math> and <math>{C}_{k}</math> are square sub-matrices of the lower, main and upper diagonal respectively.<ref>{{Cite book |last=Dietl |first=Guido K. E. |url=https://www.worldcat.org/title/ocm85898525 |title=Linear estimation and detection in Krylov subspaces |date=2007 |publisher=Springer |isbn=978-3-540-68478-7 |series=Foundations in signal processing, communications and networking |location=Berlin; New York |pages=85, 87 |language=en |oclc=ocm85898525}}</ref><ref>{{Cite book |last1=Horn |first1=Roger A. |title=Matrix analysis |last2=Johnson |first2=Charles R. |date=2017 |publisher=Cambridge University Press |isbn=978-0-521-83940-2 |edition=Second edition, corrected reprint |location=New York, NY |pages=36 |language=en}}</ref>

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<ref>{{Cite book |last=Datta |first=Biswa Nath |title=Numerical linear algebra and applications |date=2010 |publisher=SIAM |isbn=978-0-89871-685-6 |edition=2 |location=Philadelphia, Pa |pages=168}}</ref> 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=== {{See also|Triangular matrix}} An <math>n \times n</math> matrix <math>A</math> is '''upper block triangular''' (or '''block upper triangular'''<ref name=":7">{{Cite book |last=Stewart |first=Gilbert W. |title=Matrix algorithms. 2: Eigensystems |date=2001 |publisher=Soc. for Industrial and Applied Mathematics |isbn=978-0-89871-503-3 |location=Philadelphia, Pa |pages=5}}</ref>) if there are positive integers <math>n_1, \ldots, n_k</math> such that <math>n = n_1 + n_2 + \ldots + n_k</math> and <math display="block">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},</math> where the matrix <math>A_{ij}</math> is <math>n_i \times n_j</math> for all <math>i, j = 1, \ldots, k</math>.<ref name=":5">{{Cite book |last=Bernstein |first=Dennis S. |title=Matrix mathematics: theory, facts, and formulas |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14039-1 |edition=2 |location=Princeton, NJ |pages=168, 298 |language=en}}</ref><ref name=":7" /> Similarly, <math>A</math> is '''lower block triangular''' if <math display="block">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},</math> where <math>A_{ij}</math> is <math>n_i \times n_j</math> for all <math>i, j = 1, \ldots, k</math>.<ref name=":5" />

===Block Toeplitz matrices=== {{See also|Toeplitz matrix}} 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 <math>A</math> is '''block Toeplitz''' if <math>A_{(i,j)} = A_{(k,l)}</math> for all <math>k - i = l - j</math>, that is,

<math display="block">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},</math>

where <math>A_i \in \mathbb{F}^{n_i \times m_i}</math>.<ref name=":5" />

===Block Hankel matrices=== {{See also|Hankel matrix}}

A matrix <math>A</math> is '''block Hankel''' if <math>A_{(i,j)} = A_{(k,l)}</math> for all <math>i + j = k + l</math>, that is,

<math display="block">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},</math>

where <math>A_i \in \mathbb{F}^{n_i \times m_i}</math>.<ref name=":5" />

==See also== * [[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)

==Notes== {{Reflist}}

==References== {{refbegin}} *{{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}} {{refend}}

[[Category:Matrices (mathematics)]] [[Category:Sparse matrices]]