On the spectra of matrices having nonnegative sums of principal minors

The localization of the eigenvalues of matrices with nonnegative sums of principal minors is studied. The discussion is carried out in particular for P-matrices. Such matrices for which almost all the eigenvalues lie on the left half plane are constructed.     

M-matrix products having positive principal minors

Sufficient conditions are given for powers and products of M-matrices to have all principal minors positive. Several of these conditions involve directed graphs of the matrices. In particular we show that if A and B are irreducible M-matrices which have longest simple circuit of length two with A+B having no simple circuit longer than three, then the product AB has all principal minors positive.     

M-matrix products having positive principal minors

Sufficient conditions are given for powers and products of M-matrices to have all principal minors positive. Several of these conditions involve directed graphs of the matrices. In particular we show that if A and B are irreducible M-matrices which have longest simple circuit of length two with A+B having no simple circuit longer than three, then the product AB has all principal minors positive.     

On matrices having equal spectral radius and spectral norm

In this paper we characterize all nxn matrices whose spectral radius equals their spectral norm. We show that for n?3 the class of these matrices contains the normal matrices as a subclass.     

On the leading principal minors of direct products

A result is given, relating the leading principal minors of a direct product of matrices to the leading principal minors of the matrices     

On the leading principal minors of direct products

A result is given, relating the leading principal minors of a direct product of matrices to the leading principal minors of the matrices     

On matrices stochastically similar to matrices with equal diagonal elements

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS^{- 1} where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.     

On matrices stochastically similar to matrices with equal diagonal elements

A necessary and sufficient condition for a matrix to be stochastically similar to a matrix with equal diagonal elements is obtained Aand B are called Stochastically similar if B=SAS ^{? 1} where S is quasi-stochastic i.e., all row sums of .S are I. An inverse elementary divisor problem for quasi-stochastic matrices is also considered.     

Matrices with positive principal minors

A new necessary and sufficient condition is given for all principal minors of a square matrix to be positive. A special subclass of such matrices, called quasidominant matrices, is also examined.     

On normal matrices with normal principal submatrices

Let A be a normal n×n matrix. This paper discusses in detail under what conditions and in what way A can be dilated to a normal matrix of order n+1 or n+2. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 229, 1995, pp. 63–94. Translated by Kh. D. Ikramov.     

On the principal minors of a matrix with a multiple eigenvalue

The property of a Hermitian n × n matrix A that all its principal minors of order n − 1 vanish is shown to be a purely algebraic implication of the fact that the lowest two coefficients of its characteristic polynomial are zero. To prove this assertion, no information on the rank or eigenvalues of A is required. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 47–49.     

Hidden Z-matrices with positive principal minors

Let $\text{C}$ denote the class of hidden Z-matrices, i.e., $\text{M∈}\text{C}$ if and only if there exist Z-matrices X and Y such that the following two conditions are satisfied:

$\text{(}\text{M}\text{1) MX = Y,}$

$\text{(}\text{M}\text{2) r}{}^{\mathrm{T}}\text{X + s}{}^{\mathrm{T}}\text{Y > 0}\text{for some}\text{r, s ? 0.}$

Let P denote the class of real square matrices having positive principal minors. The class $\text{C}$ has arisen recently as a generalization of the class of Z-matrices [9,23,24]. In this paper, we explore various matrix-theoretic aspects of the class $\text{C}\text{∩P}$.     

On generators of bounded ratios of minors for totally positive matrices

We provide a method for factoring all bounded ratios of the form

     

On the principal components of sample covariance matrices

We introduce a class of \(M \times M\) sample covariance matrices \({\mathcal {Q}}\) which subsumes and generalizes several previous models. The associated population covariance matrix \(\Sigma = \mathbb {E}{\mathcal {Q}}\) is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of \(\Sigma - I_M\) may depend on \(M\) in an arbitrary fashion. We investigate the principal components, i.e. the top eigenvalues and eigenvectors, of \({\mathcal {Q}}\). We derive precise large deviation estimates on the generalized components \(\langle {\mathbf{{w}}} , {\varvec{\xi }_i}\rangle \) of the outlier and non-outlier eigenvectors \(\varvec{\xi }_i\). Our results also hold near the so-called BBP transition, where outliers are created or annihilated, and for degenerate or near-degenerate outliers. We believe the obtained rates of convergence to be optimal. In addition, we derive the asymptotic distribution of the generalized components of the non-outlier eigenvectors. A novel observation arising from our results is that, unlike the eigenvalues, the eigenvectors of the principal components contain information about the subcritical spikes of \(\Sigma \). The proofs use several results on the eigenvalues and eigenvectors of the uncorrelated matrix \({\mathcal {Q}}\), satisfying \(\mathbb {E}{\mathcal {Q}} = I_M\), as input: the isotropic local Marchenko–Pastur law established in Bloemendal et al. (Electron J Probab 19:1–53, 2014), level repulsion, and quantum unique ergodicity of the eigenvectors. The latter is a special case of a new universality result for the joint eigenvalue–eigenvector distribution.     

When the positivity of the leading principal minors implies the positivity of all principal minors of a matrix

An n-by-n real matrix A enjoys the “leading implies all” (LIA) property, if, whenever D   is a diagonal matrix such that A+D$A+D$ has positive leading principal minors (PMs), all PMs of A are positive. Symmetric and Z-matrices are known to have this property. We give a new class of matrices (“mixed matrices”) that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. “Nested implies all” (NIA) is also enjoyed by this new class.     

Principal minors and diagonal similarity of matrices

Let A, B be n × n matrices with entries in a field F. Our purpose is to show the following theorem: Suppose n⩾4, A is irreducible, and for every partition of {1,2,…,n} into subsets α, β with ¦α¦⩾2, ¦β¦⩾2 either rank A[α¦β]⩾2 or rank A[β¦α]⩾2. If A and B have equal corresponding principal minors, of all orders, then B or B^t is diagonally similar to A.     

Positivity of the principal minors of a nonsingularM-matrix

One gives an elementary proof of the known fact that if the principal minors of an M -matrix with a positive determinant are nonnegative, then they are positive.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, p. 88, 1981.In conclusion, we note that this problem was suggested to the author by V. P. If'in.