80 research outputs found
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. Β© 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. Β© 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 Γ 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 nΓn 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 2Γ2 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. Β© 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, 1β€p<n. For given nΓn 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.Π ΡΡΠ°ΡΡΠ΅ ΠΏΡΠΈΠ²Π΅Π΄Π΅Π½Ρ ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΡ ΠΎΡΠ΅Π½ΠΊΠΈ ΠΠ Π’ Π΄ΠΈΡΡΡΠ·ΠΈΠΈ Π² Π΄ΠΈΠ°Π³Π½ΠΎΡΡΠΈΠΊΠ΅ ΡΡΡΡΠΊΡΡΡΠ½ΡΡ
ΠΏΠΎΠ²ΡΠ΅ΠΆΠ΄Π΅Π½ΠΈΠΉ Π³ΠΎΠ»ΠΎΠ²Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ·Π³Π° Ρ Π΄Π΅ΡΠ΅ΠΉ Ρ ΡΠΈΠΌΠΏΡΠΎΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΏΠΈΠ»Π΅ΠΏΡΠΈΠ΅ΠΉ Π½Π° ΡΠΎΠ½Π΅ Π°Π½ΠΎΠΌΠ°Π»ΠΈΠΉ ΡΠ°Π·Π²ΠΈΡΠΈΡ Π³ΠΎΠ»ΠΎΠ²Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ·Π³Π° ΠΈ ΡΠΎΠΏΠΎΡΡΠ°Π²Π»Π΅Π½ΠΈΠ΅ Π²ΡΡΠ²Π»Π΅Π½Π½ΡΡ
ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠΉ Ρ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠ°ΠΌΠΈ ΡΡΡΠΈΠ½Π½ΠΎΠΉ ΠΠ Π’. ΠΠΏΠΈΡΠ°Π½Ρ ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΠΈ ΠΌΠΎΡΡΠΎΡΡΠ½ΠΊΡΠΈΠΎΠ½Π°Π»ΡΠ½ΡΡ
ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΠΉ Π²Π΅ΡΠ΅ΡΡΠ²Π° ΠΌΠΎΠ·Π³Π°, ΠΊΠΎΡΠΎΡΡΠ΅ Π΄Π°ΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡ ΠΎΡΠ΅Π½ΠΈΠ²Π°ΡΡ ΡΠ°Π·Π²ΠΈΡΠΈΠ΅ ΠΈ ΡΠ΅ΡΠ΅Π½ΠΈΠ΅ ΡΠΏΠΈΠ»Π΅ΠΏΡΠΈΠΈ Ρ Π΄Π΅ΡΠ΅ΠΉ Ρ Π°Π½ΠΎΠΌΠ°Π»ΠΈΡΠΌΠΈ ΡΠ°Π·Π²ΠΈΡΠΈΡ Π³ΠΎΠ»ΠΎΠ²Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ·Π³Π°. Π‘ΠΈΠΌΠΏΡΠΎΠΌΡ Π°Π½ΠΎΠΌΠ°Π»ΠΈΠΉ Π³ΠΎΠ»ΠΎΠ²Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ·Π³Π° Π·Π°Π²ΠΈΡΡΡ ΠΎΡ ΡΠ°ΡΠΏΠΎΠ»ΠΎΠΆΠ΅Π½ΠΈΡ ΠΈ ΡΠ°ΡΠΏΡΠΎΡΡΡΠ°Π½Π΅Π½Π½ΠΎΡΡΠΈ Π½Π°ΡΡΠ΅Π½ΠΈΠΉ
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.Π ΡΡΠ°ΡΡΠ΅ ΠΏΡΠ΅Π΄ΡΡΠ°Π²Π»Π΅Π½ Π°Π½Π°Π»ΠΈΠ· 54 MPT-ΡΠΎΠΌΠΎΠ³ΡΠ°ΠΌΠΌ ΠΈ 26 ΡΠ΅Π·ΡΠ»ΡΡΠ°ΡΠΎΠ² ΠΠ -Π΄ΠΈΡΡΡΠ·ΠΈΠΈ Ρ Π΄Π΅ΡΠ΅ΠΉ Ρ ΡΠΈΠΌΠΏΡΠΎΠΌΠ°ΡΠΈΡΠ΅ΡΠΊΠΎΠΉ ΡΠΏΠΈΠ»Π΅ΠΏΡΠΈΠΈ ΠΏΡΠΈ Π΄Π΅ΡΡΠΊΠΎΠΌ ΡΠ΅ΡΠ΅Π±ΡΠ°Π»ΡΠ½ΠΎΠΌ ΠΏΠ°ΡΠ°Π»ΠΈΡΠ΅. Π ΡΠ³Π°Π½Π½Π°Ρ ΠΠ Π’ ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½ΠΈΠ΅ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ Π²ΡΡΠ²ΠΈΡΡ ΡΡΡΡΠΊΡΡΡΠ½ΡΠ΅ Π½Π°ΡΡΡΠ΅Π½ΠΈΡ Π³ΠΎΠ»ΠΎΠ²Π½ΠΎΠ³ΠΎ ΠΌΠΎΠ·Π³Π°, ΡΠΎΡΠ° ΠΊΠ°ΠΊ ΠΠ -Π΄ΠΈΡΡΡΠ·ΠΈΡ ΡΡΡΠ°Π½Π°Π²Π»ΠΈΠ²Π°Π΅Ρ Π΄ΠΈΡΠ±Π°Π»Π°Π½Ρ Π½Π΅ΠΉΡΠΎΠ³Π΅Π½Π΅Π·Π° Ρ Π΄Π°Π½Π½ΡΡ
Π΄Π΅ΡΠ΅ΠΉ, ΠΊΠΎΡΠΎΡΡΠΉ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠΈΠ·ΡΠ΅ΡΡΡ Π΄ΠΎΡΡΠΎΠ²Π΅ΡΠ½ΡΠΌ ΠΏΠΎΠ²ΡΡΠ΅Π½ΠΈΠ΅ΠΌ ΠΏΠΎΠΊΠ°Π·Π°ΡΠ΅Π»Π΅ΠΉ ΡΡΠ΅Π΄Π½Π΅ΠΉ Π΄ΠΈΡΡΡΠ·ΠΈΠΎΠ½Π½ΠΎΠΉ ΡΠΏΠΎΡΠΎΠ±Π½ΠΎΡΡΠΈ ΠΌΠΎΠ·Π³Π° Π½Π° ΡΠΎΠ½Π΅ ΡΠ½ΠΈΠΆΠ΅Π½ΠΈΡ ΡΡΠ°ΠΊΡΠΈΠΎΠ½Π½ΠΎΠΉ Π°Π½ΠΈΠ·ΠΎΡΡΠΎΠΏΠΈΠΈ Π² Π»ΠΎΠ±Π½ΠΎΒΠ²ΠΈΡΠΎΡΠ½ΠΎΠΉ Π΄ΠΎΠ»Π΅, ΡΡΠΎ ΡΠ²ΠΈΠ΄Π΅ΡΠ΅Π»ΡΡΡΠ²ΡΠ΅Ρ ΠΎ ΠΏΡΠΎΠ½ΠΈΡΠ°Π΅ΠΌΠΎΡΡΠΈ ΠΌΠΈΠ΅Π»ΠΈΠ½ΠΎΠ²ΠΎΠΉ ΠΎΠ±ΠΎΠ»ΠΎΡΠΊΠΈ
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