In this paper, we first give the representation of the general solution of the following least-squares problem (LSP): Given matrices X ∈ Rn×p, B ∈ Rp×p and A0 ∈ Rr×r , find a matrix A ∈ Rn×n such that ‖XTAX − B‖ = min, s. t. A([1, r]) = A0, where A([1, r]) is the r×r leading principal submatrix of the matrix A. We then consider a best approximation problem: given an n× n matrix à with Ã([1, r])...

We say that X = [xi j ]n i, j=1 is centrosymmetric if xi j = xn− j+1,n−i+1, 1 i, j n. In this paper, we present an efficient algorithm for minimizing ‖AX B−C‖ where ‖·‖ is the Frobenius norm, A∈Rm×n , B∈ Rn×s , C ∈Rm×s and X ∈Rn×n is centrosymmetric with a specified central submatrix [xi j ]p i, j n−p . Our algorithm produces a suitable X such that AX B=C in finitely many steps, if such an X ex...

An important theorem in Gaussian quantum information tells us that we can diagonalise the covariance matrix of any state via a symplectic transformation. Whilst diagonal form is easy to find, process for finding diagonalising be more difficult, and common, existing method requires taking powers, which demanding analytically. Inspired by recently presented technique eigenvectors Hermitian from c...

The principal submatrix localization problem deals with recovering a K ×K principal submatrix of elevated mean μ in a large n × n symmetric matrix subject to additive standard Gaussian noise. This problem serves as a prototypical example for community detection, in which the community corresponds to the support of the submatrix. The main result of this paper is that in the regime Ω( √ n) ≤ K ≤ ...

Let A∈Rn×n be an irreducible totally nonnegative matrix with rank r and principal p, that is, all minors of A are nonnegative, is the size largest invertible square submatrix p its submatrix. triple (n,r,p) called realizable if there exists n×n p. In this work we present a method to construct matrices associated prescribed Jordan canonical form corresponding zero eigenvalue.

Matrix We take a detour to introduce abstract matrices. We want to acquire a good understanding of such matrices, since they not only represent matroids, but 74 Chapter 3. From Graphs to Matroids also display a lot of structural information about matroids that other ways do not. An abstract matrix B is a {0, 1} matrix with row and column indices plus a function called abstract determinant and d...

In this paper I present selected properties of triangular matrices and basic properties of the rank of matrices over a field. I define a submatrix as a matrix formed by selecting certain rows and columns from a bigger matrix. That is in my considerations, as an array, it is cut down to those entries constrained by row and column. Then I introduce the concept of the rank of a m× n matrix A by th...

The PolyMAX estimator is used intensively in modal analysis applications nowadays. The main advantages are its speed and the very clear stabilization diagrams it yields. Recently, an algorithm allowing a fast calculation of confidence intervals has been derived for frequency-domain least-squares estimators based on a common-denominator model. In this contribution, the approach is extended to th...

We propose a convex optimization formulation with the nuclear norm and `1-norm to find a large approximately rank-one submatrix of a given nonnegative matrix. We develop optimality conditions for the formulation and characterize the properties of the optimal solutions. We establish conditions under which the optimal solution of the convex formulation has a specific sparse structure. Finally, we...