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All about the ⊥ with its applications in the linear statistical models

Augustyn Markiewicz, Simo Puntanen (2015)

Open Mathematics

For an n x m real matrix A the matrix A⊥ is defined as a matrix spanning the orthocomplement of the column space of A, when the orthogonality is defined with respect to the standard inner product ⟨x, y⟩ = x'y. In this paper we collect together various properties of the ⊥ operation and its applications in linear statistical models. Results covering the more general inner products are also considered. We also provide a rather extensive list of references

An improvement of an inequality of Fiedler leading to a new conjecture on nonnegative matrices

Assaf Goldberger, Neumann, Michael (2004)

Czechoslovak Mathematical Journal

Suppose that A is an n × n nonnegative matrix whose eigenvalues are λ = ρ ( A ) , λ 2 , ... , λ n . Fiedler and others have shown that det ( λ I - A ) ≤ λ n - ρ n , for all λ > ρ , with equality for any such λ if and only if A is the simple cycle matrix. Let a i be the signed sum of the determinants of the principal submatrices of A of order i × i , i = 1 , ... , n - 1 . We use similar techniques to Fiedler to show that Fiedler’s inequality can be strengthened to: det ( λ I - A ) + ∑ i = 1 n - 1 ρ n - 2 i | a i | ( λ - ρ ) i ≤ λ n - ρ n , for all λ ≥ ρ . We use this inequality to derive the inequality that: ∏ 2 n ( ρ - λ i ) ≤ ρ n - 2 ∑ i = 2 n ( ρ - λ i ) . In the spirit of a celebrated conjecture due to Boyle-Handelman,...

An iterative algorithm for computing the cycle mean of a Toeplitz matrix in special form

Peter Szabó (2013)

Kybernetika

The paper presents an iterative algorithm for computing the maximum cycle mean (or eigenvalue) of n × n triangular Toeplitz matrix in max-plus algebra. The problem is solved by an iterative algorithm which is applied to special cycles. These cycles of triangular Toeplitz matrices are characterized by sub-partitions of n - 1 .

An operator-theoretic approach to truncated moment problems

Raúl Curto (1997)

Banach Center Publications

We survey recent developments in operator theory and moment problems, beginning with the study of quadratic hyponormality for unilateral weighted shifts, its connections with truncated Hamburger, Stieltjes, Hausdorff and Toeplitz moment problems, and the subsequent proof that polynomially hyponormal operators need not be subnormal. We present a general elementary approach to truncated moment problems in one or several real or complex variables, based on matrix positivity and extension. Together...

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