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In mathematics, a block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices. [1] [2]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 it, into a collection of smaller matrices.
Any n × n square matrix A whose elements are in an algebraically closed field K is similar to a Jordan matrix J, also in (), which is unique up to a permutation of its diagonal blocks themselves. J is called the Jordan normal form of A and corresponds to a generalization of the diagonalization procedure.
A square diagonal matrix is a symmetric matrix, so this can also be called a symmetric diagonal matrix. The following matrix is square diagonal matrix: [] If the entries are real numbers or complex numbers, then it is a normal matrix as well. In the remainder of this article we will consider only square diagonal matrices, and refer to them ...
The fundamental fact about diagonalizable maps and matrices is expressed by the following: An matrix over a field is diagonalizable if and only if the sum of the dimensions of its eigenspaces is equal to , which is the case if and only if there exists a basis of consisting of eigenvectors of .
In general, a square complex matrix A is similar to a block diagonal matrix = [] where each block J i is a square matrix of the form = []. So there exists an invertible matrix P such that P −1 AP = J is such that the only non-zero entries of J are on the diagonal and the superdiagonal.
Block-diagonal matrix: A block matrix with entries only on the diagonal. Block matrix: A matrix partitioned in sub-matrices called blocks. Block tridiagonal matrix: A block matrix which is essentially a tridiagonal matrix but with submatrices in place of scalar elements. Boolean matrix: A matrix whose entries are taken from a Boolean algebra ...
An circulant matrix takes the form = [] or the transpose of this form (by choice of notation). If each is a square matrix, then the matrix is called a block-circulant matrix.. A circulant matrix is fully specified by one vector, , which appears as the first column (or row) of .
In mathematics, a square matrix is said to be diagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row.