A multiplication theorem for two-variable positive real matrices

Singular value inequalities for positive semidefinite matrices

In this note‎, ‎we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique‎. ‎Our results are similar to some inequalities shown by Bhatia and Kittaneh in [Linear Algebra Appl‎. ‎308 (2000) 203-211] and [Linear Algebra Appl‎. ‎428 (2008) 2177-2191]‎.

Dirichlet’s theorem a real variable approach

Although our proof is standard (essentially the same as presented by Serre [2], whose was the first account that I understood) we eschew complexvariable theory. Instead we use simple estimates from real-variable theory. Only one new difficulty is introduced: proving that L(1, χ) 6= 0 for nontrivial real-valued Dirichlet characters χ. We use an argument of Monsky [1] to prove this fact. Our argu...

Index of Hadamard Multiplication by Positive Matrices Ii

Given a definite nonnegative matrix A ∈ Mn(C), we study the minimal index of A : I(A) = max{λ ≥ 0 : A ◦ B ≥ λB for all 0 ≤ B}, where A ◦ B denotes the Hadamard product (A ◦ B)ij = AijBij . For any unitary invariant norm N in Mn(C), we consider the Nindex of A: I(N,A) = min{N(A ◦ B) : B ≥ 0 and N(B) = 1} If A has nonnegative entries, then I(A) = I(‖ · ‖sp, A) if and only if there exists a vector...

Two modified Jacobi methods for M-matrices

Two variable orthogonal polynomials and structured matrices

We consider bivariate real valued polynomials orthogonal with respect to a positive linear functional. The lexicographical and reverse lexicographical orderings are used to order the monomials. Recurrence formulas are derived between polynomials of different degrees. These formulas link the orthogonal polynomials constructed using the lexicographical ordering with those constructed using the re...