Persymmetric Research Articles

Verified Error Bounds for Real Eigenvalues of Real Symmetric and Persymmetric Matrices

This paper mainly investigates the verification of real eigenvalues of the real symmetric and persymmetric matrices. For a real symmetric or persymmetric matrix, we use eig code in Matlab to obtain its real eigenvalues on the basis of numerical computation and provide an algorithm to compute verified error bound such that there exists a perturbation matrix of the same type within the computed error bound whose exact real eigenvalues are the computed real eigenvalues.

Nonnegative persymmetric matrices with prescribed elementary divisors

The nonnegative inverse elementary divisors problem (NIEDP) is the problem of finding conditions for the existence of an n×n entrywise nonnegative matrix A with prescribed elementary divisors. We consider the case in which the solution matrix A is required to be persymmetric. Persymmetric matrices are common in physical sciences and engineering. They arise, for instance, in the control of mechanical and electric vibrations. In this paper, we solve the NIEDP for n×n matrices assuming that (i) there exists a partition of the given list Λ={λ1,…,λn} in sublists Λk, along with suitably chosen Perron eigenvalues, which are realizable by nonnegative matrices Ak with certain of the prescribed elementary divisors, and (ii) a nonnegative persymmetric matrix exists with diagonal entries being the Perron eigenvalues of the matrices Ak, with certain of the prescribed elementary divisors. Our results generate an algorithmic procedure to compute the structured solution matrix.

Linear spaces and preservers of bounded rank-two per-symmetric triangular matrices

Let F be a field and m,n be integers m,n > 3. Let SMn(F) and STn(F) denote the linear space of n × n per-symmetric matrices over F and the linear space of n × n per-symmetric triangular matrices over F, respectively. In this note, the structure of spaces of bounded rank-two matrices of STn(F) is determined. Using this structural result, a classification of bounded rank-two linear preservers : STn(F) ! SMm(F), with F of characteristic not two, is obtained. As a corollary, a complete description of bounded rank-two linear preservers between per-symmetric triangular matrix spaces over a field of characteristic not two is addressed.

Powers of complex persymmetric or skew-persymmetric anti-tridiagonal matrices with constant anti-diagonals

In this paper, we derive a general expression for the entries of the powers of any complex persymmetric or skew-persymmetric anti-tridiagonal matrix with constant anti-diagonals, in terms of the Chebyshev polynomials of the second kind.

Powers of real persymmetric anti-tridiagonal matrices with constant anti-diagonals

In this paper, we present an eigenvalue decomposition with an orthogonal eigenvector matrix for any real persymmetric anti-tridiagonal matrix with constant anti-diagonals. Furthermore, we derive a general expression for the entries of the powers of these anti-tridiagonal matrices in terms of the Chebyshev polynomials of the second kind.

DOA Estimation With Circular Array via Spatial Averaging Algorithm

This letter presents a method to estimate the directions of arrival (DOAs) by taking spatial averaging algorithm for circular array. By utilizing the symmetric configuration of the circular array to form a centrosymmetric array, the covariance matrix of the received data can be constructed in the form of the Hermitian persymmetric matrix, and then the spatial averaging algorithm can be applied to estimate the DOAs. Compared with the conventional MUSIC method, the method proposed in this letter has a better performance at low SNR and few snapshots. Simulations are performed to demonstrate the effectiveness of our method.

Centrosymmetric isospectral flows and some inverse eigenvalue problems

This paper is concerned with the solution of some structured inverse eigenvalue problems in the class of centrosymmetric matrices. For this aim, isospectral flows evolving in the space of centrosymmetric matrices are considered to numerically construct a symmetric Toeplitz matrix or a persymmetric Hankel matrix from prescribed eigenvalues. We establish a link between the two problems and we investigate the use of simultaneously diagonalizable algebra based on sine transform [Linear Algebra Appl. 52/53 (1983) 992] to choose the starting centrosymmetric matrices for the isospectral flows. Some numerical tests show that our approach can tackle both problems when the solvability is guaranteed and it can give good insights when the existence of the solution is not guaranteed.

Inverse eigenproblem of anti-symmetric and persymmetric matrices and itsapproximation

The problem of generating a matrixA with specifiedeigenpair, where Ais an anti-symmetric and persymmetric matrix, is presented. Thesolvability conditions are studied. A general expression of such a matrixis provided. We denote the set of such matrices by \U0001d49c\U0001d4aeEn.The best approximation problem associated with \U0001d49c\U0001d4aeEnis discussed, that is: to find the nearest matrix to a given matrixA* by A ∊ \U0001d49c\U0001d4aeEn.The existence and uniqueness of the best approximation problem is proved andthe expression of this nearest matrix is provided.

Adaptive antenna array using real symmetric array covariance matrix

A novel scheme using a real symmetric array covariance matrix for an adaptive antenna array in DS/CDMA is proposed. A real symmetric array covariance matrix is estimated from a complex array covariance matrix using a unitary transformation and Toeplitz matrix approximation methods. Especially, the Hermitian Toeplitzation method not only estimates a persymmetric matrix form for a real symmetric array covariance matrix but also enhances the performance of the received signal by removing the undesired effect obtained from a complex array covariance matrix. Simulation results demonstrate the effectiveness of the proposed scheme.

Perron root bounding for nonnegative persymmetric matrices

The eigenvalue problem for a symmetric persymmetric matrix can be reduced to two symmetric eigenvalue problems of lower order. In this paper, we find in which of these problems the Perron root of a nonnegative symmetric persymmetric matrix lies. This is applied to bound the Perron root of such class of matrices.

Persymmetric determinants 5. Families of overlapping coaxial equivalent determinants

In an earlier paper the author constructed two infinite matrices and showed that they contain families of distinct submatrices whose determinants represent identical polynomials. The object of this paper is to extend the earlier results in two directions. The first set of identities is closely related to the original set and may be regarded as a supplement to the original paper. The second set relates determinants whose elements are based partly on Stirling numbers of the Second Kind to polynomials whose terms are based on Stirling numbers of the First Kind.

Persymmetric determinants

The object of this paper is to develop some of the results in the author's joint paper with Dale [2] concerning the derivatives of persymmetric determinants whose elements are Appell functions.Four new double-sum identities are presented which are valid for arbitrary persymmetric determinants. Two of these identities are applied to give direct proofs of two results in [2], A simple formula is given for the derivative of a Turanian of order n with Appell polynomial elements and the result is applied repeatedly to show that its degree is far lower than expected. It is shown that one particular determinant has simple derivatives of all orders and that its degree too is far lower than expected. The formula for the derivative of (first) cofactors is shown to be extensible in a simple manner to the derivatives of second cofactors.

Persymmetric determinants

The object of this paper is to show that three determinantal identities which appeared in the author's joint paper with Dale [3], and which at the time were thought to be independent, are in fact simple illustrations of a general family of identities. Let M 1 be a persymmetric matrix of modified Appell polynomials ψm(x) and let M 2 be a symmetric matrix of the form where A is a persymmetric array of the constants ψm(0)Pis an array of x-elements with binomial coefficients and O is an array of zeros. Then to each submatrix of M 1 there corresponds a family of distinct submatrices of M 2 the determinants of which represent identical polynomials. The result can be extended to functions of two variables. The paper ends with two determinantal representations of ψm(x) and an identity relating three determinants.

Further decomposition of the pseudoinverse and eigensystem of a Hermitian persymmetric matrix

It is known that the computation of the pseudoinverse or eigensystem of a Hermitian persymmetric matrix decomposes into the corresponding computation for a real symmetric matrix of the same order. Herein it is shown that the decomposition proceeds further if the real symmetric matrix is of special form.

A note on the generalisation of Aitken's δ2transformation

Shank's generalisation (1955) of Aitken's delta 2 transformation is presented as a ratio of persymmetric determinants, and an algorithm for calculating these is given. The superiority of the generalised transformation over repeated delta 2 transformations for summing perturbation series is shown.

On the Padé table of 𝑐𝑜𝑠𝑧

The Padé table for cos ⁡ z \cos z and related functions is written in a form that involves the binomial coefficients, the Euler numbers and Bernoulli numbers. Identities involving the corresponding persymmetric determinants and their minors are developed.

Reduction of the Pseudoinverse of a Hermitian Persymmetric Matrix

Reduction of the Pseudoinverse of a Hermitian Persymmetric Matrix

Reduction of the pseudoinverse of a Hermitian persymmetric matrix

When the pseudoinverse of a Hermitian persymmetric matrix is computed, both computer time and storage can be reduced by taking advantage of the special structure of the matrix.

Identities involving powers of persymmetric determinants

Our main result is that the fourth power of the determinant of an arbitrary m by m persymmetric matrix is identically equal to the determinant of a certain 2m by 2m skew-symmetric matrix. Having proved this, we derive several other identities involving powers of persymmetric determinants.

III.—On some Problems involving the Persymmetric Determinants

Consider a system of “orthogonal polynomials”P0, P1 (x), P2(x), …possessing the propertyx1, x2, …, xr being real given numbers, and p (x) a given “weight” function.