The problem of best approximating, a given square complex matrix in the Frobenius norm by normal matrices under a given spectral restriction is considered. The ne cessary and sufficient condition for the solvability ...The problem of best approximating, a given square complex matrix in the Frobenius norm by normal matrices under a given spectral restriction is considered. The ne cessary and sufficient condition for the solvability of the problem is given. A numerical algorithm for solving the problem is provided and a numerical example is presented.展开更多
Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are r...Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.展开更多
It is known that if A∈Mn is normal (AA*=A*A) , then AA￣=A￣A if and only if AAT=ATA. This leads to the question: do both AA￣=A￣A and AAT=ATA?imply that A?is normal? We give an example to show that this is false wh...It is known that if A∈Mn is normal (AA*=A*A) , then AA￣=A￣A if and only if AAT=ATA. This leads to the question: do both AA￣=A￣A and AAT=ATA?imply that A?is normal? We give an example to show that this is false when n=4, but we show that it is true when n=2?and n=3.展开更多
Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincaré are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these ...Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincaré are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.展开更多
The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year w...The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid CFSG or not. On the other hand, what happens for infinite groups? Since the status of knowledge of the non-commuting graph and of the prime graph is satisfactory, is it possible to find relations between these two graphs and the Sylow graph? In the present note we make the point of the situation and formulate the above questions in appropriate way.展开更多
In this paper, one construction of Cartesian authentication codes from the normal form of matrices over finite fields are presented and its size parameters are computed. Moreover, assume that the encoding rules are ch...In this paper, one construction of Cartesian authentication codes from the normal form of matrices over finite fields are presented and its size parameters are computed. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the P I and P S , which denote the largest probabilities of a successful impersonation attack and of a successful substitution attack respectively, of these codes are also computed.展开更多
文摘The problem of best approximating, a given square complex matrix in the Frobenius norm by normal matrices under a given spectral restriction is considered. The ne cessary and sufficient condition for the solvability of the problem is given. A numerical algorithm for solving the problem is provided and a numerical example is presented.
文摘Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.
文摘It is known that if A∈Mn is normal (AA*=A*A) , then AA￣=A￣A if and only if AAT=ATA. This leads to the question: do both AA￣=A￣A and AAT=ATA?imply that A?is normal? We give an example to show that this is false when n=4, but we show that it is true when n=2?and n=3.
文摘Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincaré are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.
文摘The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid CFSG or not. On the other hand, what happens for infinite groups? Since the status of knowledge of the non-commuting graph and of the prime graph is satisfactory, is it possible to find relations between these two graphs and the Sylow graph? In the present note we make the point of the situation and formulate the above questions in appropriate way.
文摘In this paper, one construction of Cartesian authentication codes from the normal form of matrices over finite fields are presented and its size parameters are computed. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the P I and P S , which denote the largest probabilities of a successful impersonation attack and of a successful substitution attack respectively, of these codes are also computed.
