Blockmatrix

In mathematics, a block matrix refers to a matrix that is interpreted as if it had been blocks into several parts, called split. A block matrix can be displayed in an intuitive manner as the original matrix with a predetermined number of horizontal and vertical dividing lines. This hyphens share the original matrix into submatrices.

Definition

Let be a matrix of size. The number of rows and columns of the matrix will now divided by and integer, and wherein the call number of summands. Then can be represented as

With sub-matrices of size. Each matrix can be interpreted in different ways as the block matrix can be divided depending on how the rows and columns. On trivial, each matrix can also be interpreted as a block matrix with only one block or as a block matrix with blocks of size.

Example

The matrix

Can be divided into four blocks

The decomposed matrix is then given by

Multiplication of block matrices

The product of block matrices can be represented purely with operations of sub-matrices. Let be a matrix with rows and columns decompositions decompositions

And a matrix with row and column decompositions decompositions

Then, the product

Can be calculated in blocks, where a matrix with rows and columns decompositions decompositions. The submatrices of the block matrix are given by

Or, using the Einstein summation convention, which summed implicitly using multiple existing indices, represented more compactly

Block diagonal matrix

A block diagonal matrix is a square block matrix whose main diagonal block matrices are square and all the remaining blocks are zero matrices. A block diagonal matrix has the form

Where the submatrices are square matrices. In other words, the direct sum of, that is,

Or with the formalism of diagonal matrices

For the determinant and the trace of a block diagonal matrix

And

The inverse of a block diagonal matrix is again a block diagonal matrix composed of the inverse of the individual blocks

The eigenvalues ​​and eigenvectors of a block diagonal matrix corresponding to the (combined ) eigenvalues ​​and eigenvectors of the submatrices.

Blocktridiagonalmatrix

A Blocktridiagonalmatrix is another special block matrix is a square matrix is exactly how the block diagonal matrix, but in addition with square block matrices in the first two ( upper and lower ) secondary diagonals. The remaining blocks are zero matrices. The Blocktridiagonalmatrix is basically a tri-diagonal matrix, but with block matrices instead of scalars. A Blocktridiagonalmatrix has the form

Being, and each square block matrices on the lower secondary diagonal, the main diagonal and upper secondary diagonal are.

Blocktridiagonalmatrizen often arise in numerical solutions of various problems (for example, in computational fluid dynamics ). There optimized numerical methods for the LU decomposition of Blocktridiagonalmatrizen and accordingly efficient method for solving systems of equations with Triadiagonalmatrizen as coefficient matrix. The Thomas algorithm, which is used for the efficient solution of systems of equations with tri-diagonal matrix can also be applied to Blocktridiagonalmatrizen.

Block - Toeplitz matrix

A block - Toeplitz matrix is another specific block matrix which, like the Toeplitz matrix containing the same blocks repeated in the diagonal. A block - Toeplitz matrix has the form