Khakim D. Ikramov

Hamiltonian square roots of skew-Hamiltonian matrices revisited

Abstract Recently, H. Fassbender et al. [Linear Algebra Appl. 287 (1999) 125] proved the following theorem: Every real skew-Hamiltonian matrix W has a real Hamiltonian square root H, i.e., H 2 =W. We prove an analog of this theorem for complex matrices. Our approach may be of independent interest, namely, we use the polar decomposition of a nonsingular operator acting in a space with the symplectic inner product.

On a QR-like Algorithm for Some StructuredEigenvalue Problems

In this paper, a QR-like algorithm, called the con-QR algorithm, for computing the Youla form of a general complex matrix is presented. The Youla form is an analog of the Schur form where unitary congruences instead of unitary similarities are employed. We introduce a set of invariants of a unitary congruence transformation which are called coneigenvalues, and discuss their condition. Finally, the practical value of the Youla form is discussed.

Unitary similarity of matrices with quadratic minimal polynomials

Abstract The fact that given complex n×n matrices A and B are (or are not) unitarily similar can be verified with the help of the Specht–Pearcy criterion. Its application, however, involves a huge amount of computational work; to get a positive answer, one should compare the traces of all the products composed of A and A ∗ with length up to 2n2 with the traces of similar products composed of B and B ∗ . For some matrices A, B, most of this work is redundant. For instance, when A and B are normal matrices, they only need to have identical eigenvalues to be unitarily similar, and this condition can be verified by comparing two n-tuples of traces. In this paper, we identify another class of matrices where unitary similarity among its members can be economically verified. These are matrices with quadratic minimal polynomials. If A and B are matrices of this kind, then, to be unitarily similar, they need to have the same eigenvalues and the same singular values, which can be verified by comparing two 2n-tuples of traces. Two widely known subclasses of matrices with quadratic minimal polynomials are projectors and involutions. For these subclasses, we give yet another derivation of the unitary similarity criterion above, based on the canonical form for oblique projectors found by D. Djokovic.

A square matrix is congruent to its transpose

For any matrix X let Xdenote its transpose. We show that if A is an n by n matrix over afi eldK ,t henA and Aare congruent over K, i.e., PAP = Afor some P ∈ GLn(K).

Some Properties of Symmetric Quasi-Definite Matrices

Symmetric quasi-definite matrices arise in numerous applications, notably in interior point methods in mathematical programming. Several authors have derived various properties of these matrices. This article provides a list of some previously known properties and adds a number of others that are believed to be new.

Computing matrix-vector products with centrosymmetric and centrohermitian matrices

An algorithm proposed recently by Melman reduces the costs of computing the product Ax with a symmetric centrosymmetric matrix A as compared to the case of an arbitrary A. We show that the same result can be achieved by a simpler algorithm, which requires only that A be centrosymmetric. However, if A is hermitian or symmetric, this can be exploited to some extent. Also, we show that similar gains are possible when A is a skew-centrosymmetric or a centrohermitian matrix.

Gaussian elimination is stable for the inverse of a diagonally dominant matrix

Let B E M n (C) be a row diagonally dominant matrix, i.e., σ i |b ii |=Σ|b ij |, i=1,...,n, where 0 ≤ σ i < 1, i = 1,...,n, with a = max 1≤i≤n σ i . We show that no pivoting is necessary when Gaussian elimination is applied to A = B -1 . Moreover, the growth factor for A does not exceed 1 + σ. The same results are true with row diagonal dominance being replaced by column diagonal dominance.

Solving the two-dimensional CIS problem by a rational algorithm

Abstract The CIS problem is formulated as follows. Let p be a fixed integer, 1⩽p . For given n×n compex matrices A and B , can one verify whether A and B have a common invariant subspace of dimension p by a procedure employing a finite number of arithmetical operations? We describe an algorithm solving the CIS problem for p=2 . Unlike the algorithm proposed earlier by the second and third authors, the new algorithm does not impose any restrictions on A and B . Moreover, when A and B generate a semisimple algebra, the algorithm is able to solve the CIS problem for any p , 1 .

Conditions for unique solvability of the matrix equation AX + X*B = C

Conditions for the unique solvability of the matrix equation AX + X*B = C are formulated in terms of the eigenvalues and the Kronecker structure of the matrix pencil A + λB* associated with this equation.

Reducibility theorems for pairs of matrices as rational criteria

Abstract Theorems giving conditions for a pair of matrices to be reducible to a special form by a simultaneous similarity transformation such as the classical McCoys theorem or theorems due to Shapiro and Watters are traditionally perceived as nonconstructive ones. We disprove this perception by showing that conditions of each of the theorems above can be verified using a finite number of arithmetic operations. A new extension of McCoys theorem is stated which, in some respects, is more convenient than Shapiros theorem.