Block matrix

Block decomposition of a (14 × 14) matrix with row and column partitions of size 2, 4 and 8

In mathematics , a designated block matrix a matrix , which is interpreted as if it were in a plurality of parts called blocks , decomposed been. A block matrix can be represented in an intuitive way as the original matrix with a certain number of horizontal and vertical dividing lines. These dividing lines divide the original matrix into sub-matrices .

definition

Be a matrix of size . The number of rows and columns of the matrix will now be divided into whole numbers using and , where and denote the number of summands. Then can be represented as ${\displaystyle \mathbf {M} }$ ${\displaystyle m\times n}$ ${\displaystyle m=m_{1}+m_{2}+\cdots +m_{q}}$ ${\displaystyle n=n_{1}+n_{2}+\cdots +n_{r}}$ ${\displaystyle q}$ ${\displaystyle r}$ ${\displaystyle \mathbf {M} }$

${\displaystyle \mathbf {M} ={\begin{bmatrix}\mathbf {M} _{11}&\mathbf {M} _{12}&\cdots &\mathbf {M} _{1r}\\\mathbf {M} _{21}&\mathbf {M} _{22}&\cdots &\mathbf {M} _{2r}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {M} _{q1}&\mathbf {M} _{q2}&\cdots &\mathbf {M} _{qr}\end{bmatrix}}}$

with sub-matrices of size . Each matrix can be interpreted as a block matrix in different ways, depending on how the rows and columns are broken down. In a trivial way, each matrix can also be understood as a block matrix with only one block or as a block matrix with blocks of the same size . ${\displaystyle \mathbf {M} _{ij}}$ ${\displaystyle m_{i}\times n_{j}}$ ${\displaystyle (m\times n)}$ ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle mn}$ ${\displaystyle 1\times 1}$

example

The matrix

${\displaystyle \mathbf {M} ={\begin{bmatrix}1&1&2&2\\1&1&2&2\\3&3&4&4\\3&3&4&4\end{bmatrix}}}$

can be broken down into four blocks ${\displaystyle (2\times 2)}$

${\displaystyle \mathbf {M} _{11}={\begin{bmatrix}1&1\\1&1\end{bmatrix}},\mathbf {M} _{12}={\begin{bmatrix}2&2\\2&2\end{bmatrix}},\mathbf {M} _{21}={\begin{bmatrix}3&3\\3&3\end{bmatrix}},\mathbf {M} _{22}={\begin{bmatrix}4&4\\4&4\end{bmatrix}}.}$

The decomposed matrix then results in

${\displaystyle \mathbf {M} ={\begin{bmatrix}\mathbf {M} _{11}&\mathbf {M} _{12}\\\mathbf {M} _{21}&\mathbf {M} _{22}\end{bmatrix}}.}$

Multiplication of block matrices

Example of a multiplication of two block matrices

The product of block matrices can be represented purely with operations on the sub-matrices. Let be a matrix with row decomposition and column decomposition ${\displaystyle \mathbf {A} }$ ${\displaystyle (m\times n)}$ ${\displaystyle q}$ ${\displaystyle r}$

${\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {A} _{11}&\mathbf {A} _{12}&\cdots &\mathbf {A} _{1r}\\\mathbf {A} _{21}&\mathbf {A} _{22}&\cdots &\mathbf {A} _{2r}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {A} _{q1}&\mathbf {A} _{q2}&\cdots &\mathbf {A} _{qr}\end{bmatrix}}}$

and a matrix with row decompositions and column decompositions ${\displaystyle \mathbf {B} }$ ${\displaystyle (n\times p)}$ ${\displaystyle r}$ ${\displaystyle s}$

${\displaystyle \mathbf {B} ={\begin{bmatrix}\mathbf {B} _{11}&\mathbf {B} _{12}&\cdots &\mathbf {B} _{1s}\\\mathbf {B} _{21}&\mathbf {B} _{22}&\cdots &\mathbf {B} _{2s}\\\vdots &\vdots &\ddots &\vdots \\\mathbf {B} _{r1}&\mathbf {B} _{r2}&\cdots &\mathbf {B} _{rs}\end{bmatrix}},}$

then applies that the product

${\displaystyle \mathbf {C} =\mathbf {A} \mathbf {B} }$

can be calculated in blocks, with a matrix with row and column decompositions . The sub-matrices of the block matrix are given by ${\displaystyle \mathbf {C} }$ ${\displaystyle (m\times p)}$ ${\displaystyle q}$ ${\displaystyle s}$ ${\displaystyle \mathbf {C} }$

${\displaystyle \mathbf {C} _{ik}=\sum _{j=1}^{r}\mathbf {A} _{ij}\mathbf {B} _{jk}.}$

Or, with the help of Einstein's summation convention , which implicitly sums over multiple indices, represented more compactly

${\displaystyle \mathbf {C} _{ik}=\mathbf {A} _{ij}\mathbf {B} _{jk}.}$

Block diagonal matrix

A block diagonal matrix is a square block matrix whose main diagonals are square block matrices and whose remaining blocks are zero matrices . A block diagonal matrix has the form ${\displaystyle \mathbf {A} }$

${\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{bmatrix}},}$

where the sub- matrices are square matrices. In other words, is the direct sum of , that is ${\displaystyle \mathbf {A} _{k}}$ ${\displaystyle \mathbf {A} }$ ${\displaystyle \mathbf {A} _{1},\dotsc ,\mathbf {A} _{n}}$

${\displaystyle \mathbf {A} =\mathbf {A} _{1}\oplus \mathbf {A} _{2}\oplus \dotsb \oplus \mathbf {A} _{n}}$

or with the formalism of diagonal matrices

${\displaystyle \mathbf {A} =\operatorname {diag} (\mathbf {A} _{1},\mathbf {A} _{2},\dotsc ,\mathbf {A} _{n})}$.

The following applies to the determinant and the trace of a block diagonal matrix

${\displaystyle \det \mathbf {A} =\det \mathbf {A} _{1}\cdot \det \mathbf {A} _{2}\dotsm \det \mathbf {A} _{n}}$

and

${\displaystyle \operatorname {Spur} (\mathbf {A} )=\operatorname {Spur} (\mathbf {A} _{1})+\dotsb +\operatorname {Spur} (\mathbf {A} _{n})}$.

The inverse of a block diagonal matrix is in turn a block diagonal matrix, composed of the inverses of the individual blocks ${\displaystyle \mathbf {A} }$

${\displaystyle {\begin{bmatrix}\mathbf {A} _{1}&0&\cdots &0\\0&\mathbf {A} _{2}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}\end{bmatrix}}^{-1}={\begin{bmatrix}\mathbf {A} _{1}^{-1}&0&\cdots &0\\0&\mathbf {A} _{2}^{-1}&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &\mathbf {A} _{n}^{-1}\end{bmatrix}}.}$

The eigenvalues and eigenvectors of a block diagonal matrix correspond to the (combined) eigenvalues ​​and eigenvectors of the sub-matrices . ${\displaystyle \mathbf {A} _{1},\mathbf {A} _{2},\dotsc ,\mathbf {A} _{n}}$

example

Important examples of block diagonal matrices are matrices in Jordan normal form . In this case, the blocks are so-called Jordan blocks, which are bidiagonal matrices with the eigenvalue of the block on the main diagonal , while all elements on the secondary diagonal are 1.

Block tridiagonal matrix

A block tridiagonal matrix is another special block matrix which, like the block diagonal matrix, is a square matrix, but with additional square block matrices in the first two (upper and lower) secondary diagonals . The remaining blocks are zero matrices. The block tridiagonal matrix is ​​basically a tridiagonal matrix , but with block matrices instead of scalars . A block tridiagonal matrix has the form ${\displaystyle \mathbf {A} }$

${\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {B} _{1}&\mathbf {C} _{1}&&&\cdots &&0\\\mathbf {A} _{2}&\mathbf {B} _{2}&\mathbf {C} _{2}&&&&\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{k}&\mathbf {B} _{k}&\mathbf {C} _{k}&&\\\vdots &&&\ddots &\ddots &\ddots &\\&&&&\mathbf {A} _{n-1}&\mathbf {B} _{n-1}&\mathbf {C} _{n-1}\\0&&\cdots &&&\mathbf {A} _{n}&\mathbf {B} _{n}\end{bmatrix}}}$

where , and are each square block matrices on the lower secondary diagonal, the main diagonal and the upper secondary diagonal. ${\displaystyle \mathbf {A} _{k}}$ ${\displaystyle \mathbf {B} _{k}}$ ${\displaystyle \mathbf {C} _{k}}$

Block tridiagonal matrices often appear in numerical solutions to various problems (for example in numerical fluid mechanics ). There are optimized numerical methods for the LR decomposition of block tridiagonal matrices and correspondingly efficient methods for solving systems of equations with triadiagonal matrices as coefficient matrix . The Thomas algorithm , which is used to efficiently solve systems of equations with a tridiagonal matrix, can also be applied to block tridiagonal matrices.

Block Toeplitz Matrix

A block Toeplitz matrix is another special block matrix which, similar to the Toeplitz matrix , contains the same blocks repeatedly on the diagonals. A block Toeplitz matrix has the form ${\displaystyle \mathbf {A} }$

${\displaystyle \mathbf {A} ={\begin{bmatrix}\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&\cdots &\mathbf {A} _{(1,n-1)}&\mathbf {A} _{(1,n)}\\\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&&&\mathbf {A} _{(1,n-1)}\\&\ddots &\ddots &\ddots &&&\vdots \\&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}&&\\\vdots &&&\ddots &\ddots &\ddots &\\\mathbf {A} _{(n-1,1)}&&&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}&\mathbf {A} _{(1,2)}\\\mathbf {A} _{(n,1)}&\mathbf {A} _{(n-1,1)}&\cdots &&&\mathbf {A} _{(2,1)}&\mathbf {A} _{(1,1)}\end{bmatrix}}.}$

See also

• Kronecker product

literature

• Gilbert Strang : Linear Algebra . Springer, Berlin a. a. 2003, ISBN 3-540-43949-8 .

Web links

• Eric W. Weisstein : Block matrix . In: MathWorld (English).
• Cam McLeman, matte: Partitioned matrix . In: PlanetMath . (English)