Editing Block matrix (section)

==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|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
:<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" />