Numerical ranges of principal submatrices

It is shown that the ratio of the area of the convex hull of the fields of values of the (n?1)-by-(n?1) principal submatrices of an n-by-n matrix A to the area of the field of values of A is bounded below by a function of n which approaches 1 as n approaches ∞. Since this convex hull is necessarily contained in the field of values of A, an interpretation is that, asymptotically in the dimension, the field of any given matrix is “filled up” by the fields of the submatrices (collectively). Some new inequalities for the eigenvalues of principal submatrices of hermitian matrices, which are not implied by interlacing, are employed.     

Normality and the numerical range

It is well known that if A is an n by n normal matrix, then the numerical range of A is the convex hull of its spectrum. The converse is valid for n ? 4 but not for larger n. In this spirit a characterization of normal matrices is given only in terms of the numerical range. Also, a characterization is given of matrices for which the numerical range coincides with the convex hull of the spectrum. A key observation is that the eigenvectors corresponding to any eigenvalue occuring on the boundary of the numerical range must be orthogonal to eigenvectors corresponding to all other eigenvalues.     

Stopping criteria for the Ando-Li-Mathias and Bini-Meini-Poloni geometric means

We provide an upper bound for the number of iterations necessary to achieve a desired level of accuracy for the Ando-Li-Mathias [Linear Algebra Appl. 385 (2004) 305-334] and Bini-Meini-Poloni [Math. Comput. 79 (2010) 437-452] symmetrization procedures for computing the geometric mean of n positive definite matrices, where accuracy is measured by the spectral norm and the Thompson metric on the convex cone of positive definite matrices. It is shown that the upper bound for the number of iterations depends only on the diameter of the set of n matrices and the desired convergence tolerance. A striking result is that the upper bound decreases as n increases on any bounded region of positive definite matrices.     

Non-skew symmetric orthogonal matrices with constant diagonals

A matrix C of order n is orthogonal if CC^T=dI. In this paper, we restrict the study to orthogonal matrices with a constant m > 1 on the diagonal and ±1's off the diagonal. It is observed that all skew symmetric orthogonal matrices of this type are constructed from skew symmetric Hadamard matrices and vice versa. Some simple necessary conditions for the existence of non-skew orthogonal matrices are derived. Two basic construction techniques for non-skew orthogonal matrices are given. Several families of non-skew orthogonal matrices are constructed by applying the basic techniques to well-known combinatorial objects like balanced incomplete block designs. It is also shown that if m is even and n=0 (mod 4), then an orthogonal matrix must be skew symmetric. The structure of a non-skew orthogonal matrix in the special case of m odd,n=2 (mod 4) and m?1/6n is also studied in detail. Finally, a list of cases with n?50 is given where the existence of non-skew orthogonal matrices are unknown.     

Three dimensional line stochastic matrices and extreme points

Several properties of the extreme points of the convex set of three dimensional line stochastic matrices of order n are presented. The existence of many different classes of extremal configurations is established. These extremal matrices exhibit a large variety of patterns with some unexpected configurations. Latin squares of special types are used in some of the existence results. Furthermore, three questions raised by Brualdi and Csima are answered concerning the extreme points of three dimensional plane stochastic matrices of order n.     

On the accuracy of the Gerschgorin circle theorem for bounding the spread of a real symmetric matrix

The spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues. The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread. For nonsymmetric matrices the Gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable. This is not true for real symmetric matrices. It is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by $\text{p+1}$, where p is the maximum number of off diagonal nonzeros in any row of the matrix. For full matrices this is just $\text{n}$. This bound is not quite sharp for n greater than 2, but examples with ratios of $\text{n?1}$ for all n are given. For banded matrices with m nonzero bands the maximum ratio is bounded by $\text{m}$ independent of the size of n. This bound is sharp provided only that n is at least 2m. For sparse matrices, p may be quite small and the Gerschgorin bound may be surprisingly accurate.     

Properties of P-sets and trapped compact convex sets

New properties of P-sets, which constitute a large class of convex compact sets in ? ⁿ that contains all convex polyhedra and strictly convex compact sets, are obtained. It is shown that the intersection of a P-set with an affine subspace is continuous in the Hausdorff metric. In this theorem, no assumption of interior nonemptiness is made, unlike in other known intersection continuity theorems for set-valued maps. It is also shown that if the graph of a set-valued map is a P-set, then this map is continuous on its entire effective set rather than only on the interior of this set. Properties of the so-called trapped sets are also studied; well-known Jung’s theorem on the existence of a minimal ball containing a given compact set in ? ⁿ is generalized. As is known, any compact set contains n + 1 (or fewer) points such that any translation by a nonzero vector takes at least one of them outside the minimal ball. This means that any compact set is trapped in the minimal ball. Compact sets trapped in any convex compact sets, rather than only in norm bodies, are considered. It is shown that, for any compact set A trapped in a P-set M ? ? ⁿ , there exists a set A ⁰ ? A trapped in M and containing at most 2n elements. An example of a convex compact set M ? ? ⁿ for which such a finite set A ⁰ ? A does not exist is given.     

Carathéodory interpolation on the non-commutative polydisk

The Carathéodory problem in the N-variable non-commutative Herglotz-Agler class and the Carathéodory-Fejér problem in the N-variable non-commutative Schur-Agler class are posed. It is shown that the Carathéodory (resp., Carathéodory-Fejér) problem has a solution if and only if the non-commutative polynomial with given operator coefficients (the data of the problem indexed by an admissible set Λ) takes operator values with positive semidefinite real part (resp., contractive operator values) on N-tuples of Λ-jointly nilpotent contractive n×n matrices, for all n∈N.     

Quasi-genera of quadratic forms

Let f be a quadratic form in n variables (n > 1) with nonzero determinant d. A prime p is said to be exceptional with respect to f if every automorph of f with rational elements, determinant ±1 and denominator prime to 2d has a denominator which is a quadratic residue of p. (Throughout, slight modifications must be made if p = 2.) Except for certain binary forms, each exceptional prime induces a splitting of the genus into two quasi-genera. Building on previous results, necessary and sufficient conditions are given that a prime p be exceptional for n = 2 and n = 3 and necessary conditions for n > 3. It is proved that there are no exceptional primes for n > 4 and only possibly in special cases for n = 4. A connection is shown between representations of integers by certain ternary forms and the existence of quasi-genera. Possible connections with spinor genera are mentioned and a few unanswered questions are posed.     

A Lie Theoretic Approach to Thompson's Theorems on Singular Values-Diagonal Elements and Some Related Results

Thompson's famous theorems on singular values–diagonalelements of the orbit of an nxn matrix A under the action (1)U(n) U(n) where A is complex, (2) SO(n) SO(n), where A isreal, (3) O(n) O(n) where A is real are fully examined. Coupledwith Kostant's result, the real semi-simple Lie algebra so_n,n yields (2) and hence (3) and the sufficient part (the hardpart) of (1). In other words, the curious subtracted term(s)are well explained. Although the diagonal elements correspondingto (1) do not form a convex set in Cⁿ, the projection of thediagonal elements into Rⁿ (or iRⁿ) is convex and the characterizationof the projection is related to weak majorization. An elementaryproof is given for this hidden convexity result. Equivalentstatements in terms of the Hadamard product are also given.The real simple Lie algebra su_{n, n} shows that such a convexityresult fits into the framework of Kostant's result. Convexityproperties and torus relations are studied. Thompson's resultson the convex hull of matrices (complex or real) with prescribedsingular values, as well as Hermitian matrices (real symmetricmatrices) with prescribed eigenvalues, are generalized in thecontext of Lie theory. Also considered are the real simple Liealgebras so_{p, q} and so_{p, q}, p < q, which yield the rectangularcases. It is proved that the real part and the imaginary partof the diagonal elements of complex symmetric matrices withprescribed singular values are identical to a convex set inRⁿ and the characterization is related to weak majorization.The convex hull of complex symmetric matrices and the convexhull of complex skew symmetric matrices with prescribed singularvalues are given. Some questions are asked.     

On the extreme points of the polytope of symmetric matrices with given row sums

The extreme points of the convex polytope of nonnegative symmetric matrices of order n with prescribed row sums are fully characterized by their respective graphs. For infinite matrices such a characterization is shown to be impossible. However, after imposing some additional conditions on the positive entries of the matrices, a considerable subfamily of infinite matrices is characterized by its graphs.     

On properties of cell matrices

In this paper properties of cell matrices are studied. A determinant of such a matrix is given in a closed form. In the proof a general method for determining a determinant of a symbolic matrix with polynomial entries, based on multivariate polynomial Lagrange interpolation, is outlined. It is shown that a cell matrix of size n>1 has exactly one positive eigenvalue. Using this result it is proven that cell matrices are (Circum-)Euclidean Distance Matrices ((C)EDM), and their generalization, k-cell matrices, are CEDM under certain natural restrictions. A characterization of k-cell matrices is outlined.     

On the optimal computation of a set of symmetric and persymmetric bilinear forms

A special class $\text{T}$ⁿ of n×n matrices is described, which has tensor rank n over the real field. A tensor base for general symmetric, persymmetric, both symmetric and persymmetric matrices and Toeplitz symmetric matrices can be defined in terms of the tensor bases of $\text{T}$^l for some different values of l. It is proved that both symmetric and persymmetric n×n matrices and Toeplitz symmetric n×n matrices have tensor rank $\text{[}\text{(n+1)}{}^2\text{4}\text{]}$ and 2n?2, respectively, in the real field.     

Approximation of convex bodies by polytopes

LetC be a convex body ofE ^d and consider the symmetric difference metric. The distance ofC to its best approximating polytope having at mostn vertices is 0 (1/n ^2/(d?1)) asn→∞. It is shown that this estimate cannot be improved for anyC of differentiability class two. These results complement analogous theorems for the Hausdorff metric. It is also shown that for both metrics the approximation properties of «most» convex bodies are rather irregular and that ford=2 «most» convex bodies have unique best approximating polygons with respect to both metrics.     

Quadratically normal matrices of type 1 and unitary similarities between them

Verification of the unitary similarity between matrices having quadratic minimal polynomials is known to be much cheaper than the use of the general Specht-Pearcy criterion. Such an economy is possible due to the following fact: n × n matrices A and B with quadratic minimal polynomials are unitarily similar if and only if A and B have the same eigenvalues and the same singular values. It is shown that this fact is also valid for a subclass of matrices with cubic minimal polynomials, namely, quadratically normal matrices of type 1.     

On the truncated assignment polytope

We investigate the convex polytope Ω_m,n(r) which is the convex hull of the m × nr-subpermutation matrices. The faces of Ω_m,n(r) are characterized, and formulae are obtained to compute their dimensions. The faces of Ω_m,n(r) are themselves convex polytopes, and we determine their facets.     

Boundedness of the gradient of a solution to the Neumann–Laplace problem in a convex domain

It is shown that solutions of the Neumann problem for the Poisson equation in an arbitrary convex n-dimensional domain are uniformly Lipschitz. Applications of this result to some aspects of regularity of solutions to the Neumann problem on convex polyhedra are given. To cite this article: V. Maz'ya, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Products of basic doubly stochastic matrices over a field

It is shown that if n?3, then every doubly stochastic matrix of order n over a field F is a product of doubly stochastic matrices with exactly two nonzero off- diagonal entries if and only if the characteristic of F is not 2 and F has more than three elements. A number of related results are also obtained.