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A. Brualdi - One of the best experts on this subject based on the ideXlab platform.
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On eigenvalues of a Rayleigh quotient matrix
1992Co-Authors: A. BrualdiAbstract:This note deals with the following problem: Let A be an n X n Hermitian matrix, and Q and 0 be two n X rn (n> m> 1) matrices both with Orthonormal Column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun. In what follows, we will consider the following interesting problem concerning the spectral variation of a Rayleigh quotient matrix: Let A be an n X n Hermitian matrix, and Q and Q be two n X m (n> m> 1) matrices both with Orthonormal Column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix Q”AQ? Here the superscript H denotes the conjugate transpose of a matrix. This problem arises often in computation methods for symmetric matrix eigenval-ues such as the power method, the QR method, the simultaneous method, and several other techniques now available. Thus it is of great importance from the point of view not only of theoretical analysis of some iterative algorithms but also of practical applications. Much work has been done for this problem in various aspects so far, e.g., [2], [S], [9], [ll], and [12]. In [S], Liu and Xu showed the following
Li Ren-cang - One of the best experts on this subject based on the ideXlab platform.
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On eigenvalues of a Rayleigh quotient matrix
Published by Elsevier Inc., 1992Co-Authors: Li Ren-cangAbstract:AbstractThis note deals with the following problem: Let A be an n × nHermitian matrix, and Q and Q̃ be two n × m (n>m⩾1) matricesboth with Orthonormal Column vectors. How do the eigenvalues of the m × m Hermitian matrix Q̃HAQ̃ differ from those of the m × m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun
Bourdon P. S. - One of the best experts on this subject based on the ideXlab platform.
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The augmented message-matrix approach to deterministic dense coding theory
'American Physical Society (APS)', 2008Co-Authors: Gerjuoy E., Williams H. T., Bourdon P. S.Abstract:A method is presented for producing analytical results applicable to the standard two-party deterministic dense coding protocol, wherein communication of K perfectly distinguishable messages is attainable with the aid of K selected local unitary operations on one qudit from a pair of entangled qudits of equal dimension d in a pure state. The method utilizes the properties of a (d^2)x(d^2) unitary matrix whose initial Columns represent message states of the system used for communication, augmented by sufficiently many additional Orthonormal Column vectors so that the resulting matrix is unitary. Using the unitarity properties of this augmented message-matrix, we produce simple proofs of previously established results including (i) an upper bound on the value of the square of the largest Schmidt coefficient, given by d/K, and (ii) the impossibility of finding a pure state that can enable transmission of K=d^2-1 messages but not d^2. Additional results obtained using the method include proofs that when K=d+1 the upper bound on the square of the largest Schmidt coefficient (i) always reduces to at least (1/2)[1+sqrt{(d-2)/(d+2)}], and (ii) reduces to (d-1)/d in the special case that the identity and shift operators are two of the selected local unitaries.Comment: 16 page
Gerjuoy E. - One of the best experts on this subject based on the ideXlab platform.
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The augmented message-matrix approach to deterministic dense coding theory
'American Physical Society (APS)', 2008Co-Authors: Gerjuoy E., Williams H. T., Bourdon P. S.Abstract:A method is presented for producing analytical results applicable to the standard two-party deterministic dense coding protocol, wherein communication of K perfectly distinguishable messages is attainable with the aid of K selected local unitary operations on one qudit from a pair of entangled qudits of equal dimension d in a pure state. The method utilizes the properties of a (d^2)x(d^2) unitary matrix whose initial Columns represent message states of the system used for communication, augmented by sufficiently many additional Orthonormal Column vectors so that the resulting matrix is unitary. Using the unitarity properties of this augmented message-matrix, we produce simple proofs of previously established results including (i) an upper bound on the value of the square of the largest Schmidt coefficient, given by d/K, and (ii) the impossibility of finding a pure state that can enable transmission of K=d^2-1 messages but not d^2. Additional results obtained using the method include proofs that when K=d+1 the upper bound on the square of the largest Schmidt coefficient (i) always reduces to at least (1/2)[1+sqrt{(d-2)/(d+2)}], and (ii) reduces to (d-1)/d in the special case that the identity and shift operators are two of the selected local unitaries.Comment: 16 page
Williams H. T. - One of the best experts on this subject based on the ideXlab platform.
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The augmented message-matrix approach to deterministic dense coding theory
'American Physical Society (APS)', 2008Co-Authors: Gerjuoy E., Williams H. T., Bourdon P. S.Abstract:A method is presented for producing analytical results applicable to the standard two-party deterministic dense coding protocol, wherein communication of K perfectly distinguishable messages is attainable with the aid of K selected local unitary operations on one qudit from a pair of entangled qudits of equal dimension d in a pure state. The method utilizes the properties of a (d^2)x(d^2) unitary matrix whose initial Columns represent message states of the system used for communication, augmented by sufficiently many additional Orthonormal Column vectors so that the resulting matrix is unitary. Using the unitarity properties of this augmented message-matrix, we produce simple proofs of previously established results including (i) an upper bound on the value of the square of the largest Schmidt coefficient, given by d/K, and (ii) the impossibility of finding a pure state that can enable transmission of K=d^2-1 messages but not d^2. Additional results obtained using the method include proofs that when K=d+1 the upper bound on the square of the largest Schmidt coefficient (i) always reduces to at least (1/2)[1+sqrt{(d-2)/(d+2)}], and (ii) reduces to (d-1)/d in the special case that the identity and shift operators are two of the selected local unitaries.Comment: 16 page
