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For every n ∈ ℕ and every field K, let M(n x n, K) be the set of n x n matrices over K, let N(n, K) be the set of nilpotent n x n matrices over K and let D(n,K) be the set of n x n matrices over K which are diagonalizable over K, that is, which are diagonalizable in M(n x n,K). Moreover, if K is a field with an involutory automorphism, let R(n,K) be the set of normal n x n matrices over K. In this short note we prove that the maximal dimension of an affine subspace in N (n, K) is n(n-1)/2 and, if the characteristic of the field is zero, an affine not linear subspace in N(n,K) has dimension less than or equal to n(n-1)/2 - 1. Moreover we prove that the maximal dimension of an affine subspace in R(n, ℂ) is n, the maximal dimension of a linear subspace in D(n, ℝ) is n(n-1)/2, while the maximal dimension of an affine not linear subspace in D(n,ℝ) is n(n-1)/2 - 1. [ABSTRACT FROM AUTHOR]

In this paper, exponential stability of continuous-time and discrete-time switched $ k\times k $ k × k block triangular systems under arbitrary switching is studied. Firstly, under the assumption that all subsystem matrices are Hurwitz and a family of those corresponding block diagonal matrices is commutative, we prove that a continuous-time switched linear system is exponentially stable under arbitrary switching. Next, under the assumption that all subsystem matrices are Hurwitz and all those block diagonal matrices are normal, it is shown that the same switched system is exponentially stable under arbitrary switching. Further, under similar conditions we prove that a discrete-time switched linear system is exponentially stable under arbitrary switching. After that, illustrative numerical examples of the obtained results are also given. Finally, we prove that $ 3\times 3 $ 3 × 3 normal matrices have nine parameter representations which are useful for numerical examples (in the Appendix). [ABSTRACT FROM AUTHOR]

A locally conformal SKT (shortly LCSKT) structure is a Hermitian structure (J , g) whose Bismut torsion 3-form H satisfies the condition d H = α ∧ H , for some closed non-zero 1-form α. This condition was introduced as a generalization of the SKT (or pluriclosed) condition d H = 0. In this paper, we characterize the almost abelian Lie algebras admitting a Hermitian structure (J , g) such that d H = α ∧ H , for some closed 1-form α. As an application we classify LCSKT almost abelian Lie algebras in dimension 6. Finally, we also study on almost abelian Lie algebras the compatibility between the LCSKT condition and other types of Hermitian structures. [ABSTRACT FROM AUTHOR]

In this short note, we presented a new proof of a weak log-majorization inequality for normal matrices and obtained a singular value inequality related to positive semi-definite matrices. What's more, we also gave an example to show that some conditions in an existing norm inequality are necessary.

The eigenstructure of imaginary axis eigenvalues of a Hamiltonian matrix is of importance in many fields of control systems, for example, in stability analysis of linear Hamiltonian systems, computation of the solutions of an algebraic Riccati equation (ARE), existence of a Lyapunov function for LTI systems, etc. The dynamical system consisting of all stationary trajectories for an optimal control problem – often called the Hamiltonian system – is known to admit, in its state space dynamical equation, a Hamiltonian matrix for the system matrix. In each of these cases, defectiveness of imaginary axis eigenvalues of a Hamiltonian matrix turns out to be of crucial importance. For example, defectiveness causes unboundedness of the oscillatory stationary trajectories in the Hamiltonian system. A characterisation of the ARE solutions in terms of Lagrangian invariant subspaces of the Hamiltonian matrix when the imaginary axis eigenvalues are defective has been addressed in the literature for the controllable case. This paper focuses on the general case of uncontrollable systems; we formulate conditions under which the imaginary axis eigenvalues of the Hamiltonian matrix are non-defective: this is central for solutions corresponding to imaginary axis eigenvalues to not become unbounded. We provide conditions on the so-called ϵ -characteristic of the non-defective imaginary axis eigenvalue: this helps in the characterisation of Lagrangian invariant subspaces. We formulate conditions under which a passivity-based Hamiltonian matrix is normal (i.e. it commutes with its transpose), and we link normality of such Hamiltonian matrices with all-pass behaviour. In summary, this paper formulates results that link defectiveness of imaginary axis eigenvalues of Hamiltonian matrix to solvabilities of Lyapunov and Algebraic Riccati equations, controllability/observability, ϵ -characteristic and sign-controllability. We consider examples in the area of bounded-real transfer functions and RLC circuits, both controllable and uncontrollable, to study applicability of the results of this paper. [ABSTRACT FROM AUTHOR]

In this article we address the stability of linear stochastic differential equations. In particular, we focus our attention on non-normality in stochastic differential equations. Following Higham and Mao we study a test problem for non-normal stochastic differential equations, that is stable without noise, and prove a property conjectured by Higham and Mao, that is that an exponentially small (in the dimension) noise term is able to destabilise in a mean-square sense the solution of the SDE. [ABSTRACT FROM AUTHOR]

Many techniques in linear algebra make use of properties of inner product spaces and orthogonality. However, when we are working with a finite field such as ..., where q is a prime power, we cannot define an inner product space since there is no way to define positivity in the finite fields. We can still define a bilinear form ... and recover many, but not all, of the techniques and results that have been used over the real and complex numbers. Using the proposed bilinear form, we define two vectors x and y to be orthogonal if = 0. We discuss the consequences of defining orthogonality in this manner, explore the existence of self-orthogonal and orthogonal bases, and establish properties of normal and Hermitian matrices. [ABSTRACT FROM AUTHOR]

Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A. The characterization of normal block Toeplitz matrices with entries from a commutative algebra A of normal matrices is also obtained. [ABSTRACT FROM AUTHOR]

Let μ 1 be a complex number in the numerical range W (A) of a normal matrix A. In the case when no eigenvalues of A lie in the interior of W (A) , we identify the smallest convex region containing all possible complex numbers μ 2 for which μ 1 ∗ 0 μ 2 is a 2-by-2 compression of A. [ABSTRACT FROM AUTHOR]

Let A ∈ C n × n be normal with eigenvalues λ 1 , ... , λ n. For a given z in the numerical range of A , consider the set B A , k (z) of k × k matrices W for which [ z ⁎ 0 W ] is a compression of A. Assuming that no three eigenvalues lie on the same line, we prove that for any compact subset K of the numerical range of A that avoids the eigenvalues, the mappings z ↦ B A , k (z) and z ↦ σ (B A , k (z)) are uniformly continuous on K. [ABSTRACT FROM AUTHOR]

We study pairs of mutually orthogonal normal matrices with respect to tropical multiplication. Minimal orthogonal pairs are characterized. The diameter and girth of three graphs arising from the orthogonality equivalence relation are computed.

Let B = J 2 n or B = R n for the matrices given by J 2 n = [ I n − I n ] ∈ M 2 n (C) or R n = [ 1 ⋰ 1 ] ∈ M n (C). A matrix A is called B -normal if A A ⋆ = A ⋆ A holds for A and its adjoint matrix A ⋆ : = B − 1 A H B. In addition, a matrix Q is called B -unitary, if Q H B Q = B. We develop sparse simple forms for nondefective (i.e. diagonalizable) J 2 n / R n -normal matrices under J 2 n / R n -unitary similarity transformations. For both cases we show that these forms exist for an open and dense subset of J 2 n / R n -normal matrices. This implies that these forms can be seen as topologically 'generic' since they exist for all J 2 n / R n -normal matrices except a nowhere dense subset. [ABSTRACT FROM AUTHOR]

Given a structured matrix A we study the problem of finding the closest normal matrix with the same structure. The structures of our interest are: Hamiltonian, skew-Hamiltonian, per-Hermitian, and perskew-Hermitian. We develop a structure-preserving Jacobi-type algorithm for finding the closest normal structured matrix and show that such algorithm converges to a stationary point of the objective function. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A set of matrices with entries from a fixed finite population P is called "Bohemian". The mnemonic comes from BOunded HEight Matrix of Integers, BOHEMI, and although the population P need not be solely made up of integers, it frequently is. In this paper we look at Bohemians, specifically those with population { − 1 , 0 , + 1 } and sometimes other populations, for instance { 0 , 1 , i , − 1 , − i }. More, we specialize the matrices to be upper Hessenberg Bohemian. We then study the characteristic polynomials of these matrices, and their height, that is the infinity norm of the vector of monomial basis coefficients. Focusing on only those matrices whose characteristic polynomials have maximal height allows us to explicitly identify these polynomials and give useful bounds on their height, and conjecture an accurate asymptotic formula. The lower bound for the maximal characteristic height is exponential in the order of the matrix; in contrast, the height of the matrices remains constant. We give theorems about the number of normal matrices and the number of stable matrices in these families. [ABSTRACT FROM AUTHOR]

Let A = {Aij}i,j∈I, where I is an index set, be a doubly indexed family of matrices, where Aij is ni × nj. For each i∈I, let Vi be an ni-dimensional vector space. We say A is reducible in the coupled sense if there exist subspaces, Ui ⊆ Vi, with Ui ≠{0} for at least one i∈I, and Ui ≠ Vi for at least one i, such that Aij(Uj) ⊆ Ui for all i, j. Let B = {Bij}i,j∈I also be a doubly indexed family of matrices, where Bij is mi times mj. For each i∈I, let Xi be a matrix of size ni times mi. Suppose Aij} Xj = Xi Bij for all i, j. We prove versions of Schur's lemma for A, mathcal B satisfying coupled irreducibility conditions. We also consider a refinement of Schur's lemma for sets of normal matrices and prove corresponding versions for A, mathcal B satisfying coupled normality and coupled irreducibility conditions. [ABSTRACT FROM AUTHOR]

In this paper we derive closed-form expressions for the coefficients of the residual and characteristic polynomials in the full orthogonalization method and GMRES iterative Krylov method for solving linear systems with diagonalizable matrices. The coefficients are given as functions of the eigenvalues and eigenvectors of the matrix A and of the right-hand side b. These results yield the residual vectors and the explicit solution of the optimization problem minp∈πk ||p(A)b||, where πk is the set of polynomials of degree k with a value 1 at the origin. In addition, the Ritz values and harmonic Ritz values can be written explicitly for the first four iterations of the Arnoldi algorithm. Moreover, from the coefficients of the characteristic polynomials, we obtain lower bounds for the distances of the eigenvalues of A to the Ritz and harmonic Ritz values. [ABSTRACT FROM AUTHOR]

We study the distance of an eigenvector of a diagonalizable matrix A to the Krylov subspace generated from A and a given starting vector v . This distance is involved in studies of the convergence of the Arnoldi method for computing eigenvalues. Contrary to the previous studies on this problem, we provide closed-form expressions for this distance in terms of the eigenvalues and eigenvectors of A as well as the components of v in the eigenvector basis. The formulas simplify when the matrix A is normal. For A non-normal we derive upper and lower bounds that are simpler than the exact expressions. We also show how to generate starting vectors such that the distance to the Krylov subspace is equal to the worst possible case. [ABSTRACT FROM AUTHOR]

This paper presents two main results that the singular values of the Hadamard product of normal matrices Ai are weakly log-majorized by the singular values of the Hadamard product of |Ai| and the singular values of the sum of normal matrices Ai are weakly log-majorized by the singular values of the sum of |Ai|. Some applications to these inequalities are also given. In addition, several related and new inequalities are obtained. [ABSTRACT FROM AUTHOR]

We study an isospectral flow (Lax flow) that provides an explicit deformation from upper Hessenberg complex matrices to normal matrices, extending to the complex case and to the case of normal matrices the results of [2] . The Lax flow is given by d A d t = [ [ A † , A ] d u , A ] , where brackets indicate the usual matrix commutator, [ A , B ] : = A B − B A , A † is the conjugate transpose of A and the matrix [ A † , A ] d u is the matrix equal to [ A † , A ] along diagonal and upper triangular entries and zero below diagonal. We prove that if the initial condition A 0 is an upper Hessenberg matrix with simple spectrum, then lim t → + ∞ ⁡ A ( t ) exists and it is a normal upper Hessenberg matrix isospectral to A 0 and if the spectrum of A 0 is contained in a line in the complex plane, then the ω -limit set is actually a tridiagonal normal matrix. Furthermore, we show that this flow is also the solution of an infinite time horizon optimal control problem and we prove that it can be used to construct even dimensional real skew-symmetric tridiagonal matrices with given simple spectrum, and with given signs pattern for the codiagonal elements. [ABSTRACT FROM AUTHOR]

We point out several implications of a result of Djoković (1971) [8] concerning orthogonality of normal matrices that satisfy a certain condition on the eigenvalues of their sum. We obtain an analogous result in the setting of conjugate-normal matrices. [ABSTRACT FROM AUTHOR]

Let J k ( λ ) be the k × k Jordan block with eigenvalue λ and let N be an m × m normal matrix. In this paper we study the polynomial numerical hulls of order 2 and n − 1 for A = J k ( λ ) ⊕ N , where n = m + k . We obtain a necessary and sufficient condition such that V 2 ( A ) has an interior point. Also, we analytically characterize V 2 ( J 2 ( λ ) ⊕ N ) and we show that if σ ( N ) ∪ { λ } is co-linear, then V 2 ( J 2 ( λ ) ⊕ N ) = ⋃ a ∈ σ ( N ) V 2 ( J 2 ( λ ) ⊕ [ a ] ) . Finally, we study V n − 1 ( A ) and we show that if σ ( N ) is neither co-linear nor co-circular, then V n − 1 ( A ) has at most one point more than σ ( A ) . [ABSTRACT FROM AUTHOR]

Given three n-tuples {λi}ni =1, {μi}ni =1, {υi}ni =1 of complex numbers, we introduce the problem of when there exists a pair of normal matrices A and B such that σ(A) = {λi}ni =1, σ(B) = {μi}ni =1, and σ(A + B) = {υi}ni =1, where σ(.) denote the spectrum. In the case when λk = 0, k = 2, . . . , n, we provide necessary and sufficient conditions for the existence of A and B. In addition, we show that the solution pair (A,B) is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when the n-tuples are real. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the problem of complete stagnation of the generalized minimum residual (GMRES) method for normal matrices. We first characterize all nonsingular normal matrices such that GMRES stagnates completely for some vector . Also we give necessary and sufficient conditions for the non-existence of a real stagnation vector for real normal matrices. The number of real stagnation vectors for normal matrices is studied. Moreover, we characterize all the eigenvalues of nonsingular normal matrices such that GMRES stagnates completely for some . Using the results derived by A. Greenbaum, V. Pták and Z. Strakoš in 1996, we consider the complete stagnation of unitary matrices and derive another characterization for all nonsingular normal matrices such that GMRES stagnates completely for some vector . [Copyright &y& Elsevier]

Abstract: For a given matrix A and right-hand side b, this paper investigates unitary matrices generating, with some right-hand sides c, the same GMRES residual norms as the pair . We give characterizations of this class of unitary matrices and point out the relationship with Krylov subspaces and Krylov residual subspaces for the pair . We investigate the eigenvalues of these unitary matrices in relation to the convergence behavior of GMRES for the pair and describe the indispensable role of the eigenvector information. We conclude with a formula for the GMRES residual norms generated by a normal matrix B in terms of its eigenvalues and components of the right-hand side c in the eigenvector basis. [Copyright &y& Elsevier]

Abstract: For several applications, it is highly desirable to understand how the eigenvalues of an imbeddable matrix , where is an isometry, are distributed throughout the numerical range of . There has been extensive study for A Hermitian, while a geometric description for the eigenvalues of imbeddings in non-Hermitian matrices remains a challenging problem. Toward this direction, a subspace is introduced, wherein all complex diagonal matrices for which a given isometry generates diagonal imbeddings are defined. In particular, conditions upon which a real diagonal matrix may be imbeddable in some normal are obtained, including an application for higher rank numerical ranges. Finally, a procedure determining whether two given sets of complex numbers may be realized as spectra of a pair of imbeddable normal matrices is established. [Copyright &y& Elsevier]

Abstract: In this note, polynomial numerical hulls of matrices of the form , where and are Hermitian, are characterized. [Copyright &y& Elsevier]

Abstract: Normal matrices in which all submatrices are normal are said to be completely normal. We characterize this class of matrices, determine the possible inertias of a particular completely normal matrix, and show that real matrices in this class are closed under (general) Schur complementation. We provide explicit formulas for the Moore–Penrose inverse of a completely normal matrix of size at least four. A result on irreducible principally normal matrices is derived as well. [Copyright &y& Elsevier]

In this note we characterize polynomial numerical hulls of matrices A ∈ Mn such that A² is Hermitian. Also, we consider normal matrices A ∈ Mn whose kth power are semidefinite. For such matrices we show that Vk (A) = σ(A). [ABSTRACT FROM AUTHOR]

Abstract: Let be a complex unital Banach algebra and let . We give regions of the complex plane which contain the spectrum of or using von Neumann spectral set theory. These results are a direct generalization of a theorem of Wielandt on the eigenvalues of the sum of two normal matrices. [Copyright &y& Elsevier]

Abstract: Let w(A) be the numerical range of a matrix and a set of points that define the spectrum σ(B) of a matrix . The problem of imbedding concerns the existence and construction of an isometry such that and is undertaken in this paper. We initially deal with hermitian matrices, for which it is well known that the necessary and sufficient condition for the imbeddability of matrix B in A is their eigenvalue interlacing, and here we present a formulation for the isometry V, when . Moreover, concerning normal matrices, a criterion for imbeddability is established in terms of the real and imaginary parts of their eigenvalues. [Copyright &y& Elsevier]

Abstract: This paper investigates the behavior of Ritz values of normal matrices. We apply Ceva’s theorem, a classical geometric result, to understand the geometric relationship between pairs of Ritz values for normal non-Hermitian matrices, and then analyze the implications for larger matrices. We find that, in the case of normal non-Hermitian matrices, the geometric constraints on the placement of Ritz values provide less freedom than the Cauchy interlacing theorem in the Hermitian case. Using our results we analyze the restarted Arnoldi method with exact shifts applied to a normal, non-Hermitian matrix. [Copyright &y& Elsevier]

Abstract: Normal matrices in which all principal submatrices are normal are said to be principally normal. Various characterizations of irreducible matrices in this class of are given. Notably, it is shown that an irreducible matrix is principally normal if and only if it is normal and all of its eigenvalues lie on a line in the complex plane. Such matrices provide a generalization of the Cauchy interlacing theorem. [Copyright &y& Elsevier]

Abstract: Let a normal matrix and a given point in its numerical range that is not an eigenvalue. In this paper, we study the problem of construction of an isometry (), such that , where the ’s do not belong to the spectrum , for . In particular, the smallest integer k is determined, such that an isometry W exists, and our approach for W is based on the construction of mutually orthogonal and A-orthogonal vectors of . [Copyright &y& Elsevier]

We obtain inequalities for the variance of complex numbers and show their connection with several old and new bounds involving the eigenval- ues of complex matrices. [ABSTRACT FROM AUTHOR]

The higher rank numerical range is closely connected to the construction of quantum error correction code for a noisy quantum channel. It is known that if a normal matrix A ∈ Mn has eigenvalue a1,…, an, then its higher rank numerical range Λk(A) is the intersection of convex polygons with vertices aji,…,ajn-k+1, where 1 ≤ j1 < ⋯ < jn-k+1 ≤ n. In this paper, it is shown that the higher rank numerical range of a normal matrix with m distinct eigenvalues can be written as the intersection of no more than max{m,4} closed half planes. In addition, given a convex polygon P, a construction is given for a normal matrix A ∈ Mn with minimum n such that Λk(A) = P. In particular, if P has p vertices, with p ≥ 3, there is a normal matrix A ∈ Mn with n ≤ max{p + k - 1, 2k + 2} such that Λk(A) = P. [ABSTRACT FROM AUTHOR]

Abstract: In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied. It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value. In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented. It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given. [Copyright &y& Elsevier]

Two theorems are proved on the condensed forms with respect to unitary similarity and congruence transformations. They provide a theoretical basis for constructing economical iterative methods for systems of linear equations whose matrices are low-rank perturbations of normal and conjugate-normal matrices. [ABSTRACT FROM AUTHOR]

Let s 1, ..., s n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s i is an eigenvalue of the leading principal submatrix A i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag( s 1, ..., s n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A i and A i + 1. [ABSTRACT FROM AUTHOR]

The method MINRES-CN was earlier proposed by the authors for solving systems of linear equations with conjugate-normal coefficient matrices. It is now shown that this method is also applicable even if the coefficient matrix, albeit not conjugate-normal, is a low-rank perturbation of a symmetric matrix. If the perturbed matrix is still conjugate-normal, then, starting from some iteration step, the recursion underlying MINRES-CN becomes a three-term relation. These results are proved in terms of matrix condensed forms with respect to unitary congruences. [ABSTRACT FROM AUTHOR]

Abstract: Let denote a finite-dimensional square complex matrix. In [L. Smithies, R.S. Varga, Singular value decomposition Geršgorin sets, J. Linear Algebra Appl. 417 (2004) 370–380; N. Fontes, J. Kover, L. Smithies, R.S. Varga, Singular value decomposition normally estimated Geršgorin sets, Electron. Trans. Numer. Anal. 26 (2007) 320–329], Professor Varga and I introduced Geršgorin-type sets which were developed from singular value decompositions (SVDs) of B. In this note, our work is extended by introducing the polar SV-Geršgorin set, . The set is a union of n closed discs in , whose centers and radii are defined in terms of the entries of a polar decomposition . The set of eigenvalues of B, , is contained in . [Copyright &y& Elsevier]

A short recurrence for orthogonalizing Krylov subspace bases for a matrix A exists if and only if the adjoint of A is a low-degree polynomial in A (i.e., A is normal of low degree). In the area of iterative methods, this result is known as the Faber-Manteuffel theorem [V. Faber and T. Manteuffel, SIAM J. Numer. Anal., 21 (1984), pp. 352-362]. Motivated by the description by J. Liesen and Z. Strako?s, we formulate here this theorem in terms of linear operators on finite dimensional Hilbert spaces and give two new proofs of the necessity part. We have chosen the linear operator rather than the matrix formulation because we found that a matrix-free proof is less technical. Of course, the linear operator result contains the Faber-Manteuffel theorem for matrices. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought. [Copyright &y& Elsevier]

We describe a rigorous analysis of unconditional stability for the alternating-direction-implicit finite-difference time-domain (ADI-FDTD) and Crank-Nicholson split step (CNSS-FDTD) schemes avoiding use of the von Neumann spectral criterion. The proof is performed in the spectral domain, and uses skew-Hermitivity of matrix terms in the AD! and CNSS total amplification matrices. A bound for the total AD! amplification matrix is provided. While the CN and CNSS-FDTD amplification matrices are unitary, the bound on the ADI-FDTD depends on the Courant number. Importantly, we have found that the ADI-FDTD amplification matrix is not normal, i.e., the unit spectral radius alone cannot be used to prove the ADI-FDTD unconditional stability. The paper also shows that the ADI-FDTD and CNSS-FDTD schemes share the same dispersion equation by a similarity of their total amplification matrices, and the two schemes have identical numerical dispersion in the frame of plane waves. [ABSTRACT FROM AUTHOR]

We show that the adjoint A+ of a matrix A with respect to a given inner product is a rational function in A, if and only if A is normal with respect to the inner product. We consider such matrices and analyze the McMillan degrees of the rational functions r such that A+ = r(A). We introduce the McMillan degree of A as the smallest among these degrees, characterize this degree in terms of the number and distribution of the eigenvalues of A, and compare the McMillan degree with the normal degree of A, which is defined as the smallest degree of a polynomial p for which A+ = p(A). We show that unless the eigenvalues of A lie on a single circle in the complex plane, the ratio of the normal degree and the McMillan degree of A is bounded by a small constant that depends neither on the number nor on the distribution of the eigenvalues of A. Our analysis is motivated by applications in the area of short recurrence Krylov subspace methods. [ABSTRACT FROM AUTHOR]