Given four complex matrices A, B, C and D where A ? Cn?n Cm?m given two distinct arbitrary numbers ?1 ?2, so that they are not eigenvalues of the matrix we find a nearest from set X to (with respect spectral norm) such B ! has prescribed ?2.

We state and prove a classical version of the Laplace expansion theorem where all submatrices in the expansion are restricted to contain a speci ed common submatrix (CSM). The result states that (accounting for signs) this restricted expansion equals the determinant of the original matrix times the determinant of the CSM. This result (Muir [Mui60], p.132) is one of many such results contained i...

We study support recovery for a $$k \times k$$ principal submatrix with elevated mean $$\lambda /N$$ , hidden in an $$N\times N$$ symmetric zero Gaussian matrix. Here >0$$ is universal constant, and we assume = N \rho $$ some constant $$\rho \in (0,1)$$ . establish that there exists $$C>0$$ such the MLE recovers proportion of if {\ge C} \sqrt{\frac{1}{\rho } \log \frac{1}{\rho }}$$ while inform...

Iterative methods are often suitable for solving least-squares problems minkAx bk2 , where A 2 R n is large and sparse. The use of the conjugate gradient method with a nonsingular square submatrix A1 2 R n of A as preconditioner was first suggested by Läuchli in 1961. This conjugate gradient method has recently been extended by Yuan to generalized least-squares problems. In this paper we consid...

The complexity, approximation and algorithmic issues of several clustering problems are studied. These non-traditional clustering problems arise from recent studies in microarray data analysis. We prove the following results. (1) Two variants of the Order-Preserving Submatrix problem are NP-hard. There are polynomial-time algorithms for the Order-Preserving Submatrix problem when the condition ...

Given an n × n matrix, its principal rank characteristic sequence is a sequence of length n + 1 of 0s and 1s where, for k = 0, 1, . . . , n, a 1 in the kth position indicates the existence of a principal submatrix of rank k and a 0 indicates the absence of such a submatrix. The principal rank characteristic sequences for symmetric matrices over various fields are investigated, with all such att...

Efficient execution of numerical algorithms requires adapting the code to the underlying execution platform. In this paper we show the process of fine tuning our sparse Hypermatrix Cholesky factorization in order to exploit efficiently two important machine resources: processor and memory. Using the techniques we presented in previous papers we tune our code on a different platform. Then, we ex...

Necessary and sufficient conditions are proven for the existence of a square matrix, over an arbitrary field, such that for every principal submatrix the sum of the elements in the row complement of the submatrix is prescribed. The problem is solved in the cases where the positions of the nonzero elements of A are contained in a given set of positions, and where there is no restriction on the p...