We study the inverses of block Toeplitz matrices based on the analysis of the block cyclic displacement. New formulas for the inverses of block Toeplitz matrices are proposed. We show that the inverses of block Toeplitz matrices can be decomposed as a sum of products of block circulant matrices. In the scalar case, the inverse formulas are proved to be numerically forward stable, if the Toeplitz matrix is nonsingular and well conditioned.
Let be an block Toeplitz matrix with blocks of size. We use the shorthand for a block Toeplitz matrix. The block Toeplitz systems arise in a variety of applications in mathematics, scientific computing, and engineering, for instance, image restoration problems in image processing, numerical differential equations and integral equations, time series analysis, and control theory [ 1 — 3 ].
If we want to solve more than one block Toeplitz linear system with the same coefficient matrix, then we usually solve four or so special block linear systems in order to determine the block Toeplitz inverse formula that expresses as the sum of products of block upper and block lower Toeplitz matrices. For example, Van Barel and Bultheel [ 4 ] gave an inverse formula for a block Toeplitz matrix and then derived a weakly stable algorithm to solve a block Toeplitz system of linear equations.
The special structure of block Toeplitz matrices has resulted in some closed formulas for their inverses. In the scalar case, Gohberg and Semencul [ 5 ] have shown that if the st entry of the inverse of a Toeplitz matrix is nonzero, then the first and the last columns of the inverse of the Toeplitz matrix are sufficient to reconstruct.
In [ 6 ], an inverse formula can be obtained by the solutions of two equations the so-called fundamental equationswhere each right-hand side of them is a shifted column of the Toeplitz matrix. As an application of NPADE, it has been shown that it can be used to compute stably, in a weak sense, the inverse of a Hankel or Toeplitz matrix.
Whenadditional problems are encountered in obtaining the inverse formula of a block Toeplitz matrix. A well-known formula of Gohberg and Heinig can constructprovided that the first and last columns together with the first and last rows of the inverse are known [ 13 ].
In [ 14 ], a set of new formulas for the inverse of a block Hankel or block Toeplitz matrix is given by Labahn et al. In [ 7 ], Ben-Artzi and Shalom have proved that each inverse of a Toeplitz matrix can be constructed via three of its columns, and thus, a parametrization of the set of inverses of Toeplitz matrices is obtained.
Then they generalized these results to block Toeplitz matrices; see [ 17 ]. In [ 18 ], Gemignani has shown that the representation of relies upon a strong structure-preserving property of the Schur complements of the nonsingular leading principal submatrices of a certain generalized Bezoutian of matrix polynomials.
In this paper, we focus our attention to the inverses of block Toeplitz matrices with the help of the block cyclic displacement. In [ 19 ], Ammar and Gader have shown that the inverse of a Toeplitz matrix can be represented as sums of products of lower triangular Toeplitz matrices and circulant matrices.
The derivation of their results is based on the idea of cyclic displacement structure. In [ 20 ], Gohberg and Olshevsky also obtained new formulas for representation of matrices and their inverses in the form of sums of products of factor circulant, which are based on the analysis of the factor cyclic displacement of matrices. Motivated by a number of related results on Toepltiz inverse formulas, we study the representation of the inverses of block Toeplitz matrices.
Since block Toeplitz matrices have similar displacement structure as Toeplitz matrices, all results about Toeplitz matrices extend quite naturally to block Toeplitz matrices. At first, for purposes of presentation, we adopt some notations that will be used throughout the paper. We denote the identity matrix by. Let be a block matrix or a block vector with block size of.
By we denote the block -circulant with the first column with a block size of ; that is, the matrix of form is defined by. In this work, new formulas for the inverses of block Toeplitz matrices are proposed. As is well known, any block Toeplitz matrix has a property of block persymmetry, that is,where is the block reverse identity matrix. Unfortunately, the matrix inverse does not hold this property. For a block Toeplitz matrixlet and be its block circulant and block skew-circulant parts, respectively.
The representation shows that there will exist block circulant matrices and such that. In this paper, by solving four linear equations with the same coefficient matrixwe decompose the inverses of block Toeplitz matrices as a sum of products of block circulant matrices.
From the point of view of the computation, the formulas involving block circulants instead of block upper or lower Toeplitz matrices are more attractive. For example, in the scalar case, matrix-vector product by a circulant matrix is roughly twice as fast as matrix-vector product by a triangular Toeplitz matrix of the same size. In fact, the efficient multiplication of a Toeplitz matrix and a vector is achieved by embedding the Toeplitz matrix in a circulant matrix of twice the size.
The derivation of the formulas we present is based on the factor cyclic displacement of block matrices. Later on, we give some corollaries about the representations of the inverse matrices.Documentation Help Center.
If the first elements of c and r differ, toeplitz issues a warning and uses the column element for the diagonal.
If r is a real vector, then r defines the first row of the matrix. If r is a complex vector with a real first element, then r defines the first row and r' defines the first column. The elements of the main diagonal are set to r 1. Create a nonsymmetric Toeplitz matrix with a specified column and row vector. Because the first elements of the column and row vectors do not match, toeplitz issues a warning and uses the column for the diagonal element.
You can create circulant matrices using toeplitz. Circulant matrices are used in applications such as circular convolution. Perform discrete-time circular convolution by using toeplitz to form the circulant matrix for convolution. Define the periodic input x and the system response h. Form the convolution matrix xConv using toeplitz. Perform discrete-time convolution by using toeplitz to form the arrays for convolution.
Define the input x and system response h. Form r by padding x with zeros. Form the column vector c. Set the first element to x 1 because the column determines the diagonal. Pad c because length c must equal length h for convolution. Column of Toeplitz matrix, specified as a scalar or vector.
If the first elements of c and r differ, toeplitz uses the column element for the diagonal. Data Types: single double int8 int16 int32 int64 uint8 uint16 uint32 uint64 Complex Number Support: Yes.
Row of Toeplitz matrix, specified as a scalar or vector. If the first elements of c and r differ, then toeplitz uses the column element for the diagonal.
A Toeplitz matrix is a diagonal-constant matrix, which means all elements along a diagonal have the same value. This function fully supports GPU arrays. This function fully supports distributed arrays.
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Click on title above or here to access this collection. An iterative procedure for the inversion of a block Toeplitz matrix is given. Hitherto published procedures are obtained as special cases of the present procedure.
The use of the procedure in time series analysis is briefly explained. Sign in Help View Cart.Partitioned Matrices or Block Matrix Multiplication
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Generating a N x N block Toeplitz Matrix out of N vectors
RSS Feeds. SIAM J. Related Databases. Web of Science You must be logged in with an active subscription to view this. Publication Data. Publisher: Society for Industrial and Applied Mathematics. Hirotugu Akaike. IEEE Access 7 Linear Models and Time-Series Analysis, Journal of Time Series Analysis 39 :3, Algorithms and Architectures for Parallel Processing, In mathematicsa block matrix or a partitioned matrix is a matrix that is interpreted as having been broken into sections called blocks or submatrices.
Block matrix algebra arises in general from biproducts in categories of matrices. It is possible to use a block partitioned matrix product that involves only algebra on submatrices of the factors. Or, using the Einstein notation that implicitly sums over repeated indices:. If a matrix is partitioned into four blocks, it can be inverted blockwise as follows:. A and D must be square, so that they can be inverted. 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.
That is, a block diagonal matrix A has the form. In other words, matrix A is the direct sum of A 1…, A n. Any square matrix can trivially be considered a block diagonal matrix with only one block. For the determinant and tracethe following properties hold.
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.
A block tridiagonal matrix is another special block matrix, which is just like the block diagonal matrix a square matrixhaving 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.
Block tridiagonal matrices are often encountered in numerical solutions of engineering problems e. 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 algorithmused 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.
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 special form of matrix transpose can also be defined for block matrices, where individual blocks are reordered but not transposed.
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 two vector spaces of matrices could be represented as a direct sum of two matrices.
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 vectors. That again matches the idea of having distinguished direct sum decompositions of the domain and 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. 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.
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Moreover, each block is a toeplitz matrix on itself. Is there a factorization for TBBT matrices analogous to the above factorization of toeplitz matricecs? These block matrices consist of circulant blocks. If we consider the blocks as elements, the BCCB matrix turns into a circulant one. The selection of the columns is done the following way. Sign up to join this community.
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Linked 1. Related 2. Hot Network Questions.Documentation Help Center. The Toeplitz block generates a Toeplitz matrix from inputs defining the first column and first row. The top input Col is a vector containing the values to be placed in the first column of the matrix, and the bottom input Row is a vector containing the values to be placed in the first row of the matrix.
The y 1,1 element is inherited from the Col input. For example, the following inputs. When you select the Symmetric check box, the block generates a symmetric Hermitian Toeplitz matrix from a single input, udefining both the first row and first column of the matrix. The output has dimension [length u length u ]. For example, the Toeplitz matrix generated from the input vector [1 2 3 4] is. When selected, enables the single-input configuration for symmetric Toeplitz matrix output.
When you generate a symmetric Toeplitz matrix with this block, if the input vector is complex, the output is a symmetric Hermitian matrix whose elements satisfy the relationship. For fixed-point signals the conjugate operation could result in an overflow.
When you select this parameter, overflows saturate. This parameter is only visible with the Symmetric parameter is selected.
This parameter is ignored for floating-point signals. Choose a web site to get translated content where available and see local events and offers.
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Toeplitz Generate matrix with Toeplitz symmetry. Description The Toeplitz block generates a Toeplitz matrix from inputs defining the first column and first row. Parameters Symmetric When selected, enables the single-input configuration for symmetric Toeplitz matrix output.
Supported Data Types Port Supported Data Types Input Double-precision floating point Single-precision floating point Fixed point signed and unsigned Boolean 8-,and bit signed integers 8-,and bit unsigned integers real signals only.In linear algebraa Toeplitz matrix or diagonal-constant matrixnamed after Otto Toeplitzis a matrix in which each descending diagonal from left to right is constant.
For instance, the following matrix is a Toeplitz matrix:. If the ij element of A is denoted A ijthen we have. A Toeplitz matrix is not necessarily square. We might therefore expect that the solution of a Toeplitz system would be easier, and indeed that is the case.
A Toeplitz matrix can also be decomposed i. Algorithms that are asymptotically faster than those of Bareiss and Levinson have been described in the literature, but their accuracy cannot be relied upon. The convolution operation can be constructed as a matrix multiplication, where one of the inputs is converted into a Toeplitz matrix. This approach can be extended to compute autocorrelationcross-correlationmoving average etc.
A bi-infinite Toeplitz matrix i.
The proof is easy to establish and can be found as Theorem 1. From Wikipedia, the free encyclopedia.
Main article: Toeplitz operator. Matrix classes. Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian.
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