Unitary similarity of algebras generated by pairs of orthoprojectors

It is shown that the unitary similarity of two matrix algebras generated by pairs of orthoprojectors { P1, Q1} and {P 2,Q2} can be verified by comparing the traces of P 1, Q1, and (P1Q1)i, i = 1, 2, n, with those of P2, Q2, and (P2Q 2)i. The conditions of the unitary similarity of two matrices with quadratic minimal polynomials presented in [A. George and Kh. D. Ikramov, Unitary similarity of matrices with quadratic minimal polynomials, Linear Algebra Appl., 349, 11-16 (2002)] are refined. Bibliography: 10 titles. © 2006 Springer Science+Business Media, Inc

A criterion for unitary congruence between complex matrices

Let A and B be square complex matrices of the same order n. Based on an important result Y. P. Hong and R. A. Horn, we propose a criterion for verifying unitary congruence of these matrices. The criterion requires that a finite number of arithmetic operations be performed. No criteria with this finiteness property were previously known. Bibliography: 7 titles. © 2012 Springer Science+Business Media, Inc

Reducibility theorems for pairs of matrices as rational criteria

Theorems giving conditions for a pair of matrices to be reducible to a special form by a simultaneous similarity transformation such as the classical McCoy's 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 McCoy's theorem is stated which, in some respects, is more convenient than Shapiro's theorem

A criterion for unitary congruence between matrices

A criteria for unitary congruence between matrices is discussed. Matrices are proved to be unitarily congruent and thus the verification of unitary congruence between two matrices reduces to the condition of unitary similarity between two matrix sets. A normal matrix family is proved to be unitarily similar to a normal matrix family and the functional is well defined if its values on the words are the same. To verify whether two families are unitarily similar, an upper bound for the length that depends only on n is required. A normal matrix family is also proved to be unitarily similar to a normal matrix family. A closed family with respect to matrix conjugation is also unitarily similar and thus a finite criterion for unitary congruence between two complex matrices is found

On the unitary similarity of matrix families

The classical Specht criterion for the unitary similarity between two complex n × n matrices is extended to the unitary similarity between two normal matrix sets of cardinality m. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht's theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht-Pearcy criterion

On condensed forms for partially commuting matrices

Two complex n×n matrices A and B are said to be partially commuting if A and B have a common eigenvector. We propose a condensed form for such matrices that can be obtained from A and B by a finite rational computation. The condensed form is a pair of block triangular matrices, with the sizes of the blocks being uniquely defined by the original matrices. We then show how to obtain additional zeros inside the diagonal blocks of a condensed form by using the generalized Lanczos procedure as given by Elsner and Ikramov. This procedure can also be considered as a finite rational process. We point out several applications of the constructions above. It turns out that for Laffey pairs of matrices, i.e., for matrices (A,B) such that rank[A,B]=1, the condensed form is a pair of 2×2 block triangular matrices. Using this fact, we show an economical way to find a spanning set for the matrix algebra generated by Laffey matrices A and B. Another application concerns so-called k-self-adjoint matrices. We examine such matrices in the unitary space as well as in a Krein space of defect 1. As an Appendix, we give a new description of the Shemesh subspace of matrices A and B. This is the maximal common invariant subspace of A and B, on which these matrices commute. © 2000 Elsevier Science Inc

Solving the two-dimensional CIS problem by a rational algorithm

The CIS problem is formulated as follows. Let p be a fixed integer, 1≤p<n. 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<p<n

Role of diffusion-weighted magnetic resonance imaging in diagnostics of symptomatic epilepsy with anomalies of brain

The paper presents the results ot the evaluation with MRI diffusion in the diagnosis of structural brain damage in children with symptomatic epilepsy in the background of developmental abnormalities of the brain and comparison of changes detected with routine MRI parameters. The features of morphological and functional changes in the brain substance, which allow to evaluate the development and course of epilepsy in children with congenital malformations of the brain. Syptoms of brain anomalies depend on the location and extent of violations.В статье приведены результаты оценки МРТ диффузии в диагностике структурных повреждений головного мозга у детей с симптоматической эпилепсией на фоне аномалий развития головного мозга и сопоставление выявленных изменений с параметрами рутинной МРТ. Описаны особенности морфофункциональных изменений вещества мозга, которые дают возможность оценивать развитие и течение эпилепсии у детей с аномалиями развития головного мозга. Симптомы аномалий головного мозга зависят от расположения и распространенности наррений

Specific indicators of MR diffusion in child cerebral palsy with symptomatic epilepsy

The article presents an analysis of 54 MR tomograms and 18 results diffusion weighted imaging children with symptomatic epilepsy with cerebral palsy. Routine MRI reveals structural disorders of the brain and the epileptogenic focus. Based on DW MRI proved imbalance neurogenesis in these children, which is characterized by a significant increase in the average performance of diffusing capacity of the brain due to lower fractional anisotropy in the fronto-temporal lobe, indicating that the permeability of the myelin sheath.В статье представлен анализ 54 MPT-томограмм и 26 результатов МР-диффузии у детей с симптоматической эпилепсии при детском церебральном параличе. Руганная МРТ исследование позволяет выявить структурные нарушения головного мозга, тоща как МР-диффузия устанавливает дисбаланс нейрогенеза у данных детей, который характеризуется достоверным повышением показателей средней диффузионной способности мозга на фоне снижения фракционной анизотропии в лобно­височной доле, что свидетельствует о проницаемости миелиновой оболочки