The almost semimonotone matrices

Megan Wendler

doi:10.1515/spma-2019-0023

The almost semimonotone matrices

Special Matrices ◽

10.1515/spma-2019-0023 ◽

2019 ◽

Vol 7 (1) ◽

pp. 291-303

Author(s):

Megan Wendler

Keyword(s):

Nonzero Vector ◽

Principal Submatrix ◽

Principal Submatrices

Abstract A (strictly) semimonotone matrix A ∈ ℝn×n is such that for every nonzero vector x ∈ ℝn with nonnegative entries, there is an index k such that xk > 0 and (Ax)k is nonnegative (positive). A matrix which is (strictly) semimonotone has the property that every principal submatrix is also (strictly) semimonotone. Thus, it becomes natural to examine the almost (strictly) semimonotone matrices which are those matrices which are not (strictly) semimonotone but whose proper principal submatrices are (strictly) semimonotone. We characterize the 2 × 2 and 3 × 3 almost (strictly) semimonotone matrices and describe many of their properties. Then we explore general almost (strictly) semimonotone matrices, including the problem of detection and construction. Finally, we relate (strict) central matrices to semimonotone matrices.

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Some algorithms for maximum volume and cross approximation of symmetric semidefinite matrices

BIT Numerical Mathematics ◽

10.1007/s10543-021-00872-1 ◽

2021 ◽

Author(s):

Stefano Massei

Keyword(s):

Computer Science ◽

Recent Work ◽

Linear Algebra ◽

Optimization Problems ◽

Approximation Error ◽

Positive Semidefinite ◽

Numerical Linear Algebra ◽

Maximum Volume ◽

Deterministic Algorithms ◽

Principal Submatrix

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

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