Unitary Similarity Transformations Research Articles

Jordan–Schur Algorithms for Computing the Matrix Exponential

In this paper, two new versions of the Schur algorithm for computing the matrix exponential of an n × n complex matrix A are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of A. The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off‐diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order n4. It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.

Matrix regularization for tensor fields

Abstract We propose a novel matrix regularization for tensor fields. In this regularization, tensor fields are described as rectangular matrices, and area-preserving diffeomorphisms and local rotations of the orthonormal frame are both realized as unitary similarity transformations of matrices in a unified way. We also show that the matrix commutator corresponds to the covariantized Poisson bracket for tensor fields in the large-N limit.

The operator system of Toeplitz matrices

A recent paper of A. Connes and W.D. van Suijlekom [Comm. Math. Phys. 383 (2021), pp. 2021–2067] identifies the operator system of n × n n\times n Toeplitz matrices with the dual of the space of all trigonometric polynomials of degree less than n n . The present paper examines this identification in somewhat more detail by showing explicitly that the Connes–van Suijlekom isomorphism is a unital complete order isomorphism of operator systems. Applications include two special results in matrix analysis: (i) that every positive linear map of the n × n n\times n complex matrices is completely positive when restricted to the operator subsystem of Toeplitz matrices and (ii) that every linear unital isometry of the n × n n\times n Toeplitz matrices into the algebra of all n × n n\times n complex matrices is a unitary similarity transformation. An operator systems approach to Toeplitz matrices yields new insights into the positivity of block Toeplitz matrices, which are viewed herein as elements of tensor product spaces of an arbitrary operator system with the operator system of n × n n\times n complex Toeplitz matrices. In particular, it is shown that min and max positivity are distinct if the blocks themselves are Toeplitz matrices, and that the maximally entangled Toeplitz matrix ξ n \xi _n generates an extremal ray in the cone of all continuous n × n n\times n Toeplitz-matrix valued functions f f on the unit circle S 1 S^1 whose Fourier coefficients f ^ ( k ) \hat f(k) vanish for | k | ≥ n |k|\geq n . Lastly, it is noted that all positive Toeplitz matrices over nuclear C ∗ ^* -algebras are approximately separable.

Generation of orthogonal rational functions by procedures for structured matrices

The problem of computing recurrence coefficients of sequences of rational functions orthogonal with respect to a discrete inner product is formulated as an inverse eigenvalue problem for a pencil of Hessenberg matrices. Two procedures are proposed to solve this inverse eigenvalue problem, via the rational Arnoldi iteration and via an updating procedure using unitary similarity transformations. The latter is shown to be numerically stable. This problem and both procedures are generalized by considering biorthogonal rational functions with respect to a bilinear form. This leads to an inverse eigenvalue problem for a pencil of tridiagonal matrices. A tridiagonal pencil implies short recurrence relations for the biorthogonal rational functions, which is more efficient than the orthogonal case. However the procedures solving this problem must rely on nonunitary operations and might not be numerically stable.

Spectral theory for self-adjoint quadratic eigenvalue problems - a review

Many physical problems require the spectral analysis of quadratic matrix polynomials $M\lambda^2+D\lambda +K$, $\lambda \in \mathbb{C}$, with $n \times n$ Hermitian matrix coefficients, $M,\;D,\;K$. In this largely expository paper, we present and discuss canonical forms for these polynomials under the action of both congruence and similarity transformations of a linearization and also $\lambda$-dependent unitary similarity transformations of the polynomial itself. Canonical structures for these processes are clarified, with no restrictions on eigenvalue multiplicities. Thus, we bring together two lines of attack: (a) analytic via direct reduction of the $n \times n$ system itself by $\lambda$-dependent unitary similarity and (b) algebraic via reduction of $2n \times 2n$ symmetric linearizations of the system by either congruence (Section 4) or similarity (Sections 5 and 6) transformations which are independent of the parameter $\lambda$. Some new results are brought to light in the process. Complete descriptions of associated canonical structures (over $\mathbb{R}$ and over $\mathbb{C}$) are provided -- including the two cases of real symmetric coefficients and complex Hermitian coefficients. These canonical structures include the so-called sign characteristic. This notion appears in the literature with different meanings depending on the choice of canonical form. These sign characteristics are studied here and connections between them are clarified. In particular, we consider which of the linearizations reproduce the (intrinsic) signs associated with the analytic (Rellich) theory (Sections 7 and 9).

A tale of three diagonalizations

In addition to the diagonalization of a normal matrix by a unitary similarity transformation, there are two other types of diagonalization procedures that sometimes arise in quantum theory applications — the singular value decomposition and the Autonne–Takagi factorization. In this pedagogical review, each of these diagonalization procedures is performed for the most general [Formula: see text] matrices for which the corresponding diagonalization is possible, and explicit analytical results are provided in each of the three cases.

On normal and structured matrices under unitary structure-preserving transformations

Structured canonical forms under unitary and suitable structure-preserving similarity transformations for normal and (skew-)Hamiltonian as well as normal and per(skew)-Hermitian matrices are proposed. Moreover, an algorithm for computing those canonical forms is sketched.

On the Existence of Schur-like Forms for Matrices with Symmetry Structures

Schur-like forms are developed for matrices that have a symmetry structure with respect to an indefinite inner product induced by a Hermitian and unitary Gram matrix. It is characterized under which conditions these forms can be computed by structure-preserving unitary transformations. The main results combines and generalizes the two well-known results from the literature that on the one hand any normal matrix can be unitarily diagonalized and on the other hand a Hamiltonian matrix can be transformed to Hamiltonian Schur form via a unitary similarity transformation if and only if its purely imaginary eigenvalues satisfy certain conditions that involve the sign characteristic of the matrix under consideration.

Look-ahead in the two-sided reduction to compact band forms for symmetric eigenvalue problems and the SVD

We address the reduction to compact band forms, via unitary similarity transformations, for the solution of symmetric eigenvalue problems and the computation of the singular value decomposition (SVD). Concretely, in the first case, we revisit the reduction to symmetric band form, while, for the second case, we propose a similar alternative, which transforms the original matrix to (unsymmetric) band form, replacing the conventional reduction method that produces a triangular–band output. In both cases, we describe algorithmic variants of the standard Level 3 Basic Linear Algebra Subroutines (BLAS)-based procedures, enhanced with look-ahead, to overcome the performance bottleneck imposed by the panel factorization. Furthermore, our solutions employ an algorithmic block size that differs from the target bandwidth, illustrating the important performance benefits of this decision. Finally, we show that our alternative compact band form for the SVD is key to introduce an effective look-ahead strategy into the corresponding reduction procedure.

Decomposition of the Nodal Conductance Matrix of a Planar Resistive Grid and Derivation of Its Eigenvalues and Eigenvectors Using the Kronecker Product and Sum With Application to CNN Image Filters

It is shown that an (M × N)-node planar resistive grid can be decomposed into two sub-grids; one made up of M N-node horizontal and the other of N M-node vertical linear resistive grids which corresponds to decomposing its nodal conductance matrix (NCM) into the Kronecker sum of the NCMs of horizontal and vertical linear grids. This enables the analytical expressions of the eigenvalues and eigenvectors of the NCMs of the sub-grids as well as those of the planar resistive grid to be expressed in terms of those of the two linear grids, whose analytical expressions are well known. For a Cellular Neural Network (CNN) Gabor-type filter (GTF) we define generalized nodal conductance matrices (GNCMs) that correspond to the NCMs of the resistive sub-grids, show that each Kronecker decomposition has a counterpart in CNN GTF and prove that each GNCM, its counterpart NCM and the corresponding temporal state matrices are related through unitary diagonal similarity transformations. Consequently, we prove that the eigenvalues of the temporal state matrix of a spatial band-pass CNN GTF are the same as those of its counterpart spatial low-pass CNN image filter, hence their temporal transient behaviors are similar in settling to a forced response.

An extended Hessenberg form for Hamiltonian matrices

A unitary symplectic similarity transformation for a special class of Hamiltonian matrices to extended Hamiltonian Hessenberg form is presented. Whereas the classical Hessenberg form links to Krylov subspaces, the extended Hessenberg form links to extended Krylov subspaces. The presented algorithm generalizes thus the classic reduction to Hamiltonian Hessenberg form and offers more freedom in the choice of Hamiltonian condensed forms, to be used within an extended Hamiltonian QR algorithm. Theoretical results identifying the structure of the extended Hamiltonian Hessenberg form and proofs of uniqueness of the reduction process are included. Numerical experiments confirm the validity of the approach.

Isometries of the spaces of self-adjoint traceless operators

In this paper we describe the structure of all isometries of the space of self-adjoint traceless operators on a finite dimensional Hilbert space under the metrics coming from the operator norm or the Schatten norms. We prove that up to a translation and multiplication by −1, they are unitary or antiunitary similarity transformations.

Toeplitz-plus-Hankel circulants are reducible to block diagonal form via unitary congruences

A Hankel circulant is a matrix obtained by reversing the order of columns (or rows) in a conventional circulant. A Toeplitz-plus-Hankel circulant (briefly, ()-circulant) is the sum of a circulant and a Hankel circulant. Bozzo discovered that the set of ()-circulants is the centralizer of the matrix , where is the cyclic permutation matrix. As a consequence, all the matrices in can be simultaneously brought to a block diagonal form with diagonal blocks of orders one and two by a unitary similarity transformation. We show that the same assertion holds for if unitary similarities are replaced by unitary congruences.

Inverse eigenvalue problems for extended Hessenberg and extended tridiagonal matrices

In inverse eigenvalue problems one tries to reconstruct a matrix, satisfying some constraints, given some spectral information. Here, two inverse eigenvalue problems are solved.First, given the eigenvalues and the first components of the associated eigenvectors (called the weight vector) an extended Hessenberg matrix with prescribed poles is computed possessing these eigenvalues and satisfying the eigenvector constraints. The extended Hessenberg matrix is retrieved by executing particularly designed unitary similarity transformations on the diagonal matrix containing the eigenvalues. This inverse problem closely links to orthogonal rational functions: the extended Hessenberg matrix contains the recurrence coefficients given the nodes (eigenvalues), poles (poles of the extended Hessenberg matrix), and a weight vector (first eigenvector components) determining the discrete inner product. Moreover, it is also sort of the inverse of the (rational) Arnoldi algorithm: instead of using the (rational) Arnoldi method to compute a Krylov basis to approximate the spectrum, we will reconstruct the orthogonal Krylov basis given the spectral info.In the second inverse eigenvalue problem, we do the same, but refrain from unitarity. As a result we execute possibly non-unitary similarity transformations on the diagonal matrix of eigenvalues to retrieve a (non)-symmetric extended tridiagonal matrix. The algorithm will be less stable, but it will be faster, as the extended tridiagonal matrix admits a low cost factorization of O(n) (n equals the number of eigenvalues), whereas the extended Hessenberg matrix does not. Again there is a close link with orthogonal function theory, the extended tridiagonal matrix captures the recurrence coefficients of bi-orthogonal rational functions. Moreover, it is again sort of inverse of the nonsymmetric Lanczos algorithm: given spectral properties, we reconstruct the two basis Krylov matrices linked to the nonsymmetric Lanczos algorithm.

On the Use of Lie Group Homomorphisms for Treating Similarity Transformations in Nonadiabatic Photochemistry

A formulation based on Lie group homomorphisms is presented for simplifying the treatment of unitary similarity transformations of Hamiltonian matrices in nonadiabatic photochemistry. A general derivation is provided whereby it is shown that a similarity transformation acting on a traceless, Hermitian matrix through a unitary matrix ofSU(n)is equivalent to the product of a single matrix ofOn2-1by a real vector. We recall how Pauli matrices are the adequate tool whenn=2and show how the same is achieved forn=3with Gell-Mann matrices.

Effective algorithms for decomplexifying a matrix by unitary similarities or congruences

It is required to verify whether a given complex n × n matrix A can be made real by a similarity or a congruence transformation. Algorithms for solving these two problems are proposed and justified under the additional assumption that A is irreducible in the former case and \(A_L = \bar AA\) is irreducible in the latter case. The irreducibility of a square complex matrix means that no unitary similarity transformation converts this matrix into a direct sum of smaller matrices. The proposed algorithms are effective in the sense that their implementation requires a finite number of arithmetic operations.

Simultaneous decomplexification of a pair of complex matrices via a unitary similarity transformation

Let (A, B) be a given pair of complex n × n matrices and let at least one of these matrices be unitarily irreducible. An algorithm for verifying whether A and B can be made real via the same unitary transformation is proposed and justified. Bibliograhy: 5 titles.

A unification of unitary similarity transforms to compressed representations

In this paper a new framework for transforming arbitrary matrices to compressed representations is presented. The framework provides a generic way of transforming a matrix via unitary similarity transformations to, e.g., Hessenberg, Hessenberg-like form and combinations of both. The new algorithms are deduced, based on the QR-factorization of the original matrix. Relying on manipulations with rotations, all the algorithms consist of eliminating the correct set of rotations, resulting in a matrix obeying the desired structural constraints. Based on this new reduction procedure we investigate further correspondences such as irreducibility, uniqueness of the reduction procedure and the link with (rational) Krylov methods. The unitary similarity transform to Hessenberg-like form as presented here, differs significantly from the one presented in earlier work. Not only does it use less rotations to obtain the desired structure, also the convergence to rational Ritz-values is not observed in the conventional approach.

On complex matrices with simple spectrum that are unitarily similar to real matrices

Suppose that one should verify whether a given complex n × n matrix can be converted into a real matrix by a unitary similarity transformation. Sufficient conditions for this property to hold were found in an earlier publication of this author. These conditions are relaxed in the following way: as before, the spectrum is required to be simple, but pairs of complex conjugate eigenvalues \( \lambda ,\bar \lambda \) are now allowed. However, the eigenvectors corresponding to such eigenvalues must not be orthogonal.

Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm

The $QR$-algorithm is a renowned method for computing all eigenvalues of an arbitrary matrix. A preliminary unitary similarity transformation to Hessenberg form is indispensable for keeping the computational complexity of the subsequent $QR$-steps under control. When restraining computing time is the vital issue, we observe that the prominent role played by the Hessenberg matrix is sufficient but perhaps not necessary to fulfill this goal. In this paper, a whole new family of matrices, sharing the major qualities of Hessenberg matrices, will be put forward. This gives rise to the development of innovative implicit $QR$-type algorithms, pursuing rotations instead of bulges. The key idea is to benefit from the $QR$-factorization of the matrices involved. The prescribed order of rotations in the decomposition of the $Q$-factor uniquely characterizes several matrix types such as Hessenberg, inverse Hessenberg, and $CMV$ matrices. Loosening the fixed ordering of these rotations provides us the class of matrices under consideration. Establishing a new implicit $QR$-type algorithm for these matrices requires a generalization of diverse well-established concepts. We consider the preliminary unitary similarity transformation, a proof of uniqueness of this reduction, an extension of the $CMV$-decomposition to a double Hessenberg factorization, and an explicit and implicit $QR$-type algorithm. A detailed complexity analysis illustrates the competitiveness of the novel method with the traditional Hessenberg approach. The numerical experiments show comparable accuracy for a wide variety of matrix types, but disclose an intriguing difference between the average number of iterations before deflation can be applied.