Abstract
In this paper, we consider a constrained low rank approximation problem: , where E is a given complex matrix, p is a positive integer, and is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation . We discuss the range of p and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.
Highlights
Throughout, let m×n denote the set of all complex m × n matrices, (n) the set of all n × n unitary matrices, and (n) the set of all n × n Hermitian matrices, + 0 ( n )the set of Hermitian nonnegative-definite matrices
Ω is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation AXA∗ = B
We discuss the range of p and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions
Summary
Throughout, let m×n denote the set of all complex m × n matrices, (n) the set of all n × n unitary matrices, and (n) the set of all n × n Hermitian matrices,. Zhang and Cheng [16], Wei and Wang [17] studied the fixed rank Hermitian nonnegative definite solution to the matrix equation AXA* = B and the least squares problem AXA* = B in Frobenius norm, which discussed the ranges of the rank k and derived expressions of the solutions by applying the SVD of the matrix of A. Motivated by the above work, we in this paper study the constrained low rank approximation of Hermitian nonnegative definite matrix.
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