366 research outputs found
Positive Definite Solutions of the Nonlinear Matrix Equation
This paper is concerned with the positive definite solutions to the matrix
equation where is the unknown and is
a given complex matrix. By introducing and studying a matrix operator on
complex matrices, it is shown that the existence of positive definite solutions
of this class of nonlinear matrix equations is equivalent to the existence of
positive definite solutions of the nonlinear matrix equation
which has been extensively studied in the
literature, where is a real matrix and is uniquely determined by It is
also shown that if the considered nonlinear matrix equation has a positive
definite solution, then it has the maximal and minimal solutions. Bounds of the
positive definite solutions are also established in terms of matrix .
Finally some sufficient conditions and necessary conditions for the existence
of positive definite solutions of the equations are also proposed
A theory of linear estimation
Theory of linear estimation and applicability to problems of smoothing, filtering, extrapolation, and nonlinear estimatio
Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum
In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel-Mickey and the Chan-Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as \mtx{X^\adj X}, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties
Quantum Systems and Alternative Unitary Descriptions
Motivated by the existence of bi-Hamiltonian classical systems and the
correspondence principle, in this paper we analyze the problem of finding
Hermitian scalar products which turn a given flow on a Hilbert space into a
unitary one. We show how different invariant Hermitian scalar products give
rise to different descriptions of a quantum system in the Ehrenfest and
Heisenberg picture.Comment: 18 page
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