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In [[mathematics]], a '''block matrix''' or a '''partitioned matrix''' is a [[matrix (mathematics)|matrix]] which is ''interpreted'' as having been broken into sections called '''blocks''' or '''submatrices'''.<ref>{{cite book |last=Eves |first=Howard |authorlink=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=37 |url=http://books.google.com/books?id=ayVxeUNbZRAC&lpg=PA40&dq=block%20multiplication&pg=PA37#v=onepage&q&f=false |edition=reprint |accessdate=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 out, 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>rowgroups</math>, and then partitioning <math>m</math> into a collection <math>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 and onto]] way to some <math>(s,t)</math> [[Offset (computer science)|offset]] entry of some <math>(x,y)</math>, where <math>x \in rowgroups</math> and <math>y \in colgroups</math>.


==Example==
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[[File:BlockMatrix168square.png|thumb |A 168×168 element block matrix with 12×12, 12×24, and 24×24 sub-Matrices. Non-zero elements are in blue, zero elements are grayed.]]
 
The matrix
 
:<math>\mathbf{P} = \begin{bmatrix}
1 & 1 & 2 & 2\\
1 & 1 & 2 & 2\\
3 & 3 & 4 & 4\\
3 & 3 & 4 & 4\end{bmatrix}</math>
 
can be partitioned into 4 2×2 blocks
 
:<math>\mathbf{P}_{11} = \begin{bmatrix}
1 & 1 \\
1 & 1 \end{bmatrix},  \mathbf{P}_{12} = \begin{bmatrix}
2 & 2\\
2 & 2\end{bmatrix},  \mathbf{P}_{21} = \begin{bmatrix}
3 & 3 \\
3 & 3 \end{bmatrix},  \mathbf{P}_{22} = \begin{bmatrix}
4 & 4\\
4 & 4\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==
A block partitioned matrix product can sometimes be used that involves only algebra on submatrices of the factors. The partitioning of the factors is not arbitrary, however, and requires "conformable partitions"<ref>{{cite book |last=Eves |first=Howard |authorlink=Howard Eves |title=Elementary Matrix Theory |year=1980 |publisher=Dover |location=New York |isbn=0-486-63946-0 |page=37 |url=http://books.google.com/books?id=ayVxeUNbZRAC&lpg=PA40&dq=block%20multiplication&pg=PA39#v=onepage&q&f=false |edition=reprint |accessdate=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 formed 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 your matrix <math>\mathbf{C}</math> are calculated by multiplying:
 
:<math>
\mathbf{C}_{\alpha \beta} = \sum^s_{\gamma=1}\mathbf{A}_{\alpha \gamma}\mathbf{B}_{\gamma \beta}.  
</math>
 
Or, using the [[Einstein notation]] that implicitly sums over repeated indices:
 
:<math>
\mathbf{C}_{\alpha \beta} = \mathbf{A}_{\alpha \gamma}\mathbf{B}_{\gamma \beta}.  
</math>
 
==Block diagonal matrices {{anchor|Block diagonal matrix}} ==
A '''block diagonal matrix''' is a block matrix which is a [[square matrix]], and having [[main diagonal]] blocks square matrices, such that the off-diagonal blocks are zero matrices.  A block diagonal matrix '''A''' has the form
 
:<math>
\mathbf{A} = \begin{bmatrix}
\mathbf{A}_{1} & 0 & \cdots & 0 \\ 0 & \mathbf{A}_{2} & \cdots &  0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \mathbf{A}_{n}
\end{bmatrix}
</math>
 
where '''A'''<sub>''k''</sub> is a square matrix; in other words, it 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;<math>\oplus</math>&nbsp;'''A'''<sub>2</sub>&nbsp;<math>\oplus\,\ldots\,\oplus </math>&nbsp;'''A'''<sub>n</sub> &nbsp;or&nbsp; diag('''A'''<sub>1</sub>, '''A'''<sub>2</sub>,<math>\ldots</math>, '''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> \operatorname{det} \mathbf{A} = \operatorname{det} \mathbf{A}_1 \times \ldots \times \operatorname{det} \mathbf{A}_n</math>,
:<math> \operatorname{tr} \mathbf{A} = \operatorname{tr} \mathbf{A}_1 +\cdots +\operatorname{tr} \mathbf{A}_n.</math>
 
The inverse of a block diagonal matrix is another block diagonal matrix, composed of the inverse of each block, as follows:
:<math>\begin{pmatrix}
\mathbf{A}_{1} & 0 & \cdots & 0 \\
0 & \mathbf{A}_{2} & \cdots &  0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \mathbf{A}_{n}
\end{pmatrix}^{-1} = \begin{pmatrix} \mathbf{A}_{1}^{-1} & 0 & \cdots & 0 \\
0 & \mathbf{A}_{2}^{-1} & \cdots &  0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \mathbf{A}_{n}^{-1}
\end{pmatrix}.
</math>
 
The eigenvalues and eigenvectors of <math>A</math> are simply those of <math>A_{1}</math> and <math>A_{2}</math> and ... and <math>A_{n}</math> (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  &        & 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}  \\
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. The individual block matrix elements, Aij, must also be a Toeplitz matrix.
 
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>
 
==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 & \cdots & \vdots & \vdots & \cdots & \vdots \\
    a_{m 1} & \cdots & a_{mn} &      0 & \cdots &      0 \\
          0 & \cdots &      0 & b_{11} & \cdots &  b_{1q} \\
    \vdots & \cdots & \vdots & \vdots & \cdots & \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.
 
==Direct Product==
{{main|Kronecker product}}
 
==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 (mathematics)|domain]] and [[range (mathematics)|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.
 
==References==
{{Reflist}}
*{{Cite web |last=Strang |first=Gilbert |authorlink=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}}
 
{{DEFAULTSORT:Block Matrix}}
[[Category:Matrices]]
[[Category:Sparse matrices]]

Revision as of 11:54, 15 February 2014


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