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

Various 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 ) .

The interesting spectral interlacing property for a certain tridiagonal matrix

In this paper, a new tridiagonal matrix, whose eigenvalues are the same as the Sylvester-Kac matrix of the same order, is provided. The interest of this matrix relies also in that the spectrum of a principal submatrix is also of a Sylvester-Kac matrix given rise to an interesting spectral interlacing property. It is proved alternatively that the initial matrix is similar to the Sylvester-Kac matrix.

Eigenvalue rigidity for truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an [Formula: see text] principal submatrix of an [Formula: see text] random unitary matrix distributed according to Haar measure. For [Formula: see text] and [Formula: see text] large with [Formula: see text], the empirical spectral measure is well approximated by a deterministic measure [Formula: see text] supported on the unit disc. In earlier work, we showed that for fixed [Formula: see text] and [Formula: see text], the bounded-Lipschitz distance [Formula: see text] between the empirical spectral measure and the corresponding [Formula: see text] is typically of order [Formula: see text] or smaller. In this paper, we consider eigenvalues on a microscopic scale, proving concentration inequalities for the eigenvalue counting function and for individual bulk eigenvalues.

Knot invariants from Laplacian matrices

A checkerboard graph of a special diagram of an oriented link is made a directed, edge-weighted graph in a natural way so that a principal submatrix of its Laplacian matrix is a Seifert matrix of the link. Doubling and weighting the edges of the graph produces a second Laplacian matrix such that a principal submatrix is an Alexander matrix of the link. The Goeritz matrix and signature invariants are obtained in a similar way. A device introduced by Kauffman makes it possible to apply the method to general diagrams.

The almost semimonotone matrices

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.

The singular acyclic matrices of even order with a P-set of maximum size

Let mA(0) denote the nullity of a given n-by-n symmetric matrix A. Set A(?) for the principal submatrix of A obtained after deleting the rows and columns indexed by the nonempty subset ? of {1,...,n}. When mA(?)(0) = mA(0) + |?|, we call ? a P-set of A. The maximum size of a P-set of A is denoted by Ps(A). It is known that Ps(A) ? ?n/2? and this bound is not sharp for singular acyclic matrices of even order. In this paper, we find the bound for this case and classify all of the underlying trees. Some illustrative examples are provided.

On the multiplicities of eigenvalues of graphs and their vertex deleted subgraphs: old and new results

Given a simple graph G, let A(G) be its adjacency matrix. A principal submatrix of A(G) of order one less than the order of G is the adjacency matrix of its vertex deleted subgraph. It is well-known that the multiplicity of any eigenvalue of A(G) and such a principal submatrix can differ by at most one. Therefore, a vertex v of G is a downer vertex (neutral vertex, or Parter vertex) with respect to a fixed eigenvalue μ if the multiplicity of μ in A(G)−v goes down by one (resp., remains the same, or goes up by one). In this paper, we consider the problems of characterizing these three types of vertices under various constraints imposed on graphs being considered, on vertices being chosen and on eigenvalues being observed. By assigning weights to edges of graphs, we generalizeour results to weighted graphs, or equivalently to symmetric matrices.