Negative powers and the spectrum of matrices
Nested matrices and inverse -matrices
Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the - and -factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse -matrices with symmetric, irreducible, tridiagonal inverses.
Netted matrices.
New bounds for the minimum eigenvalue of 𝓜-tensors
A new lower bound and a new upper bound for the minimum eigenvalue of an 𝓜-tensor are obtained. It is proved that the new lower and upper bounds improve the corresponding bounds provided by He and Huang (J. Inequal. Appl., 2014, 2014, 114) and Zhao and Sang (J. Inequal. Appl., 2016, 2016, 268). Finally, two numerical examples are given to verify the theoretical results.
New bounds for the minimum eigenvalue of the Fan product of two -matrices
In this paper, we mainly use the properties of the minimum eigenvalue of the Fan product of -matrices and Cauchy-Schwarz inequality, and propose some new bounds for the minimum eigenvalue of the Fan product of two -matrices. These results involve the maximum absolute value of off-diagonal entries of each row. Hence, the lower bounds for the minimum eigenvalue are easily calculated in the practical examples. In theory, a comparison is given in this paper. Finally, to illustrate our results, a simple...
New bounds for the minimum eigenvalue ofM-matrices
Some new bounds for the minimum eigenvalue of M-matrices are obtained. These inequalities improve existing results, and the estimating formulas are easier to calculate since they only depend on the entries of matrices. Finally, some examples are also given to show that the bounds are better than some previous results.
New criteria for H-tensors and an application
Some new criteria for identifying H-tensors are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor are given. Advantages of results obtained are illustrated by numerical examples.
New estimates for the solution of the Lyapunov matrix differential equation.
New families of integer matrices whose leading principal minors form some well-known sequences.
New iterative codes for𝓗-tensors and an application
New iterative codes for identifying 𝓗 -tensor are obtained. As an application, some sufficient conditions of the positive definiteness for an even-order real symmetric tensor, i.e., an even-degree homogeneous polynomial form are given. Advantages of results obtained are illustrated by numerical examples.
New kinds of theorems on non-negative matrices
New light on the theorem of Perron.
We prove that the principal eigenvector of a positive matrix represents the relative dominance of its rows or ranking of alternatives in a decision represented by the rows of a pairwise comparison matrix.
New limit theorems related to free multiplicative convolution
Let ⊞, ⊠, and ⊎ be the free additive, free multiplicative, and boolean additive convolutions, respectively. For a probability measure μ on [0,∞) with finite second moment, we find a scaling limit of as N goes to infinity. The -transform of its limit distribution can be represented by Lambert’s W-function. From this, we deduce that the limiting distribution is freely infinitely divisible, like the lognormal distribution in the classical case. We also show a similar limit theorem by replacing free...
New lower solution bounds for the continuous algebraic Riccati equation.
New representations for the Moore-Penrose inverse.
New results about semi-positive matrices
Our purpose is to present a number of new facts about the structure of semipositive matrices, involving patterns, spectra and Jordon form, sums and products, and matrix equivalence, etc. Techniques used to obtain the results may be of independent interest. Examples include: any matrix with at least two columns is a sum, and any matrix with at least two rows, a product, of semipositive matrices. Any spectrum of a real matrix with at least elements is the spectrum of a square semipositive matrix,...
New results for EP matrices in indefinite inner product spaces
In this paper we study -EP matrices, as a generalization of EP-matrices in indefinite inner product spaces, with respect to indefinite matrix product. We give some properties concerning EP and -EP matrices and find connection between them. Also, we present some results for reverse order law for Moore-Penrose inverse in indefinite setting. Finally, we deal with the star partial ordering and improve some results given in the “EP matrices in indefinite inner product spaces” (2012), by relaxing some...
New scaling of Itzykson–Zuber integrals
New SOR-like methods for solving the Sylvester equation
We present new iterative methods for solving the Sylvester equation belonging to the class of SOR-like methods, based on the SOR (Successive Over-Relaxation) method for solving linear systems. We discuss convergence characteristics of the methods. Numerical experimentation results are included, illustrating the theoretical results and some other noteworthy properties of the Methods.
New stability conditions for positive continuous-discrete 2D linear systems
New necessary and sufficient conditions for asymptotic stability of positive continuous-discrete 2D linear systems are established. Necessary conditions for the stability are also given. The stability tests are demonstrated on numerical examples.