scholarly journals Refinements of Kantorovich Inequality for Hermitian Matrices

2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Feixiang Chen
Keyword(s):  
Hermitian Matrix ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Hermitian Matrices ◽  
Kantorovich Inequality

Some new Kantorovich-type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be invertible and provides refinements of the classical results.

The Eigenvalues of Complementary Principal Submatrices of a Positive Definite Matrix

1972 ◽  
Vol 24 (4) ◽  
pp. 658-667 ◽  
Author(s):  
R. C. Thompson ◽  
S. Therianos
Keyword(s):  
Hermitian Matrix ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Principal Submatrices ◽  
Image Position

Let C be an n-square Hermitian matrix, presented in partitioned form aswhere A is a-square and B is b-square. Let denote the eigenvalues of C, A, B, respectively. In a recent paper [10] the following inequality was established:1.1if1.2This inequality is a simplification and a sharpening of an inequality established earlier in [6], and is a wide generalization of an inequality of Aronszajn [4].

Some inequalities concerning positive-definite Hermitian matrices

1955 ◽  
Vol 51 (3) ◽  
pp. 414-421 ◽  
Author(s):  
Ky Fan
Keyword(s):  
Hermitian Matrix ◽  
Positive Definite ◽  
Hermitian Matrices ◽  
Principal Submatrix ◽  
Image Position

1. Let H = (aij) be a positive-definite Hermitian matrix of order n. For any k distinct integers i1, i2, …, ik between 1 and n, we shall use the symbol (i1, i2, …, ik) to denote the k-rowed principal submatrix of H corresponding to the rows and columns with indices i1, i2, …, ik. It is well known thatMand more generally,

scholarly journals Computing Hermitian determinantal representations of hyperbolic curves

2015 ◽  
Vol 25 (08) ◽  
pp. 1327-1336 ◽  
Author(s):  
Daniel Plaumann ◽  
Rainer Sinn ◽  
David E. Speyer ◽  
Cynthia Vinzant
Keyword(s):  
Linear Algebra ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Numerical Implementation ◽  
Hermitian Matrices ◽  
Hyperbolic Form ◽  
Computationally Intensive

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation.

scholarly journals A note on majorization properties of the Lieb function

2020 ◽  
Vol 36 (36) ◽  
pp. 134-142
Author(s):  
Marek Niezgoda
Keyword(s):  
Hermitian Matrix ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Doubly Stochastic ◽  
Stochastic Operator ◽  
Commuting Matrices ◽  
Doubly Stochastic Operator ◽  
Block Diagonal

In this note, the Lieb function $(A,B) \to \Phi (A,B) = \tr \exp ( A + \log B )$ for an Hermitian matrix $A$ and a positive definite matrix $B$ is studied. It is shown that $\Phi$ satisfies a majorization property of Sherman type induced by a doubly stochastic operator. The variant for commuting matrices is also considered. An interpretation is given for the case of the orthoprojection operator onto the space of block diagonal matrices.

scholarly journals A Matrix Inequality for the Inversions of the Restrictions of a Positive Definite Hermitian Matrix

2013 ◽  
Vol 03 (04) ◽  
pp. 55-58 ◽  
Author(s):  
Weixiong Mai ◽  
Mo Yan ◽  
Tao Qian ◽  
Matteo Dalla Riva ◽  
Saburou Saitoh
Keyword(s):  
Hermitian Matrix ◽  
Positive Definite ◽  
Matrix Inequality

scholarly journals A geometric statement of the Harnack inequality for a degenerate Kolmogorov equation with rough coefficients

2019 ◽  
Vol 21 (07) ◽  
pp. 1850057 ◽  
Author(s):  
Francesca Anceschi ◽  
Michela Eleuteri ◽  
Sergio Polidoro
Keyword(s):  
Maximum Principle ◽  
Harnack Inequality ◽  
Weak Solutions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Strong Maximum Principle ◽  
Kolmogorov Equation ◽  
Measurable Coefficients ◽  
Rough Coefficients ◽  
Vector Formula

We consider weak solutions of second-order partial differential equations of Kolmogorov–Fokker–Planck-type with measurable coefficients in the form [Formula: see text] where [Formula: see text] is a symmetric uniformly positive definite matrix with bounded measurable coefficients; [Formula: see text] and the components of the vector [Formula: see text] are bounded and measurable functions. We give a geometric statement of the Harnack inequality recently proved by Golse et al. As a corollary, we obtain a strong maximum principle.

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