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{{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 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.

:<math>
\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>\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>\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>
  \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><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>

==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>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>\sum_{i=1}^{p} m_i = m</math>, and <math>\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>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>A = \begin{bmatrix}
a_1^T \\
a_2^T \\
\vdots \\
a_m^T
\end{bmatrix}</math>,

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

=== Common partitions ===
Often,<ref name=":2" /> we encounter the 2x2 partition

:<math>A = \begin{bmatrix}
A_{11} & A_{12} \\
A_{21} & A_{22}
\end{bmatrix}</math>,<ref name=":2" />

particularly in the form where <math>A_{11}</math> is a scalar:

:<math>A = \begin{bmatrix}
a_{11} & a_{12}^T \\
a_{21} & A_{22}
\end{bmatrix}</math>.<ref name=":2" />
==Block matrix operations==

===Transpose===
Let

:<math>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}^{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

:<math>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>,<ref name=":2" /><ref name=":1" />

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>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

:<math>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>.<ref name=":2" />

===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

:<math>
  C = AB
</math>

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:

:<math>
  C_{ij} = \sum_{k=1}^{q} A_{ik}B_{kj}. 
</math><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:

:<math>
  C_{ij} = A_{ik}B_{kj}. 
</math>

Depicting <math>C</math> as a matrix, we have

:<math>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>.<ref name=":2" />

===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>{P} = \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}^{-1} = \begin{bmatrix}
     {A}^{-1} + {A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
      -{A}^{-1}{B}\left({D} - {CA}^{-1}{B}\right)^{-1} \\
    -\left({D}-{CA}^{-1}{B}\right)^{-1}{CA}^{-1} &
       \left({D} - {CA}^{-1}{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>{P} = \begin{bmatrix}
    {A} & {B} \\
    {C} & {D}
  \end{bmatrix}^{-1} = \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} &
       \quad {D}^{-1} + {D}^{-1}{C}\left({A} - {BD}^{-1}{C}\right)^{-1}{BD}^{-1}
  \end{bmatrix}.
</math><ref name=":0" />

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}
    {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.

===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{bmatrix}A& 0\\ C& D\end{bmatrix} = \det(A) \det(D) = \det\begin{bmatrix}A& B\\ 0& D\end{bmatrix}.</math><ref name=":0" />

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

:<math>\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(A) \det\left(D - C A^{-1} B\right),</math><ref name=":0" />
and if <math>D</math> is invertible, one has

:<math>\det\begin{bmatrix}A& B\\ C& D\end{bmatrix} = \det(D) \det\left(A - B D^{-1} C\right) .</math><ref>Taboga, Marco (2021). "Determinant of a block matrix", Lectures on matrix algebra.</ref><ref name=":0" />

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{bmatrix}A& B\\ C& D\end{bmatrix} = \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 is also true when <math>AB=BA</math>, <math>AC=CA</math>, or {{tmath|1=BD=DB}}.
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)
:<math>\det\begin{bmatrix}A& B\\ B& A\end{bmatrix} = \det(A - B) \det(A + B).</math><ref name=":0" />

==Special types of block matrices==

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

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

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.

====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>{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'''<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>.<ref name=":0" /> It can also be indicated as '''A'''<sub>1</sub>&nbsp;⊕&nbsp;'''A'''<sub>2</sub>&nbsp;⊕&nbsp;...&nbsp;⊕&nbsp;'''A'''<sub>''n''</sub><ref name=":1" /> or diag('''A'''<sub>1</sub>, '''A'''<sub>2</sub>, ..., '''A'''<sub>''n''</sub>)<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>&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{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}}</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
:<math>\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><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>

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>{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}}
====Upper block triangular====
A 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

:<math>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 <math>A_{ij} \in \mathbb{F}^{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" />

====Lower block triangular====
A matrix <math>A</math> is '''lower block triangular''' if

:<math>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} \in \mathbb{F}^{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>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>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==
*{{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]]