scholarly journals A special property of the matrix Riccati equation

1974 ◽  
Vol 10 (2) ◽  
pp. 245-253 ◽  
Author(s):  
A.N. Stokes
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Symmetric Matrix ◽  
Special Property ◽  
Positive Definiteness ◽  
Symmetric Matrices ◽  
Unusual Property ◽  
Riccati Differential Equation ◽  
The Matrix ◽  
Independent Variable

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.

scholarly journals Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Ondřej Došlý ◽  
Hana Funková
Keyword(s):  
Differential Equation ◽  
Riccati Equation ◽  
Riccati Differential Equation ◽  
Oscillatory Properties

We investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.

scholarly journals A canonical form and solution for the matrix Riccati differential equation

1985 ◽  
Vol 26 (3) ◽  
pp. 355-361 ◽  
Author(s):  
R. B. Leipnik
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Canonical Form ◽  
The Self ◽  
State Equations ◽  
Riccati Differential Equation ◽  
Adjoint Matrix ◽  
Transient Problem ◽  
The Matrix ◽  
Constant Coefficients

AbstractA canonical form of the self-adjoint Matrix Riccati Differential Equation with constant coefficients is obtained in terms of extremal solutions of the self-adjoint Matrix Riccati Algebraic (steady-state) Equations. This form is exploited in order to obtain a convenient explicit solution of the transient problem. Estimates of the convergence rate to the steady state are derived.

scholarly journals The enhanced principal rank characteristic sequence over a field of characteristic 2

2017 ◽  
Vol 32 ◽  
pp. 273-290 ◽  
Author(s):  
Xavier Martínez-Rivera
Keyword(s):  
Symmetric Matrix ◽  
Complete Characterization ◽  
Symmetric Matrices ◽  
Characteristic Sequence ◽  
The Matrix ◽  
Characteristic 2

The enhanced principal rank characteristic sequence (epr-sequence) of an $n \times n$ symmetric matrix over a field $\F$ was recently defined as $\ell_1 \ell_2 \cdots \ell_n$, where $\ell_k$ is either $\tt A$, $\tt S$, or $\tt N$ based on whether all, some (but not all), or none of the order-$k$ principal minors of the matrix are nonzero. Here, a complete characterization of the epr-sequences that are attainable by symmetric matrices over the field $\Z_2$, the integers modulo $2$, is established. Contrary to the attainable epr-sequences over a field of characteristic $0$, this characterization reveals that the attainable epr-sequences over $\Z_2$ possess very special structures. For more general fields of characteristic $2$, some restrictions on attainable epr-sequences are obtained.

scholarly journals Accessibility of the matrix Riccati differential equation

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 4130031-4130032
Author(s):  
G. Dirr ◽  
U. Helmke
Keyword(s):  
Differential Equation ◽  
Riccati Differential Equation ◽  
The Matrix

scholarly journals Orthogonal and symplectic Harish-Chandra integrals and matrix product ensembles

2019 ◽  
Vol 08 (04) ◽  
pp. 1950015 ◽  
Author(s):  
Peter J. Forrester ◽  
Jesper R. Ipsen ◽  
Dang-Zheng Liu ◽  
Lun Zhang
Keyword(s):  
Probability Density Function ◽  
Diffusion Equation ◽  
Laplace Transform ◽  
Symmetric Matrix ◽  
Singular Values ◽  
Symmetric Matrices ◽  
Matrix Product ◽  
The Matrix ◽  
Random Product ◽  
Structure Formula

In this paper, we highlight the role played by orthogonal and symplectic Harish-Chandra integrals in the study of real-valued matrix product ensembles. By making use of these integrals and the matrix-valued Fourier-Laplace transform, we find the explicit eigenvalue distributions for particular Hermitian anti-symmetric matrices and Hermitian anti-self dual matrices, involving both sums and products. As a consequence of these results, the eigenvalue probability density function of the random product structure [Formula: see text], where each [Formula: see text] is a standard real Gaussian matrix, and [Formula: see text] is a real anti-symmetric matrix can be determined. For [Formula: see text] and [Formula: see text] the bidiagonal anti-symmetric matrix with 1’s above the diagonal, this reclaims results of Defosseux. For general [Formula: see text], and this choice of [Formula: see text], or [Formula: see text] itself a standard Gaussian anti-symmetric matrix, the eigenvalue distribution is shown to coincide with that of the squared singular values for the product of certain complex Gaussian matrices first studied by Akemann et al. As a point of independent interest, we also include a self-contained diffusion equation derivation of the orthogonal and symplectic Harish-Chandra integrals.

scholarly journals Sufficient conditions for symmetric matrices to have exactly one positive eigenvalue

2020 ◽  
Vol 8 (1) ◽  
pp. 98-103
Author(s):  
Doaa Al-Saafin ◽  
Jürgen Garloff
Keyword(s):  
Symmetric Matrix ◽  
Sufficient Conditions ◽  
Positive Eigenvalue ◽  
Bernstein Function ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Real Symmetric Matrix ◽  
The Matrix

AbstractLet A = [aij] be a real symmetric matrix. If f : (0, ∞) → [0, ∞) is a Bernstein function, a sufficient condition for the matrix [f (aij)] to have only one positive eigenvalue is presented. By using this result, new results for a symmetric matrix with exactly one positive eigenvalue, e.g., properties of its Hadamard powers, are derived.

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