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Block diagonalization

Jaromír J. Koliha (2001)

Mathematica Bohemica

We study block diagonalization of matrices induced by resolutions of the unit matrix into the sum of idempotent matrices. We show that the block diagonal matrices have disjoint spectra if and only if each idempotent matrix in the inducing resolution double commutes with the given matrix. Applications include a new characterization of an eigenprojection and of the Drazin inverse of a given matrix.

Block Factorization of Hankel Matrices and Euclidean Algorithm

S. Belhaj (2010)

Mathematical Modelling of Natural Phenomena

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < ...

Block matrix approximation via entropy loss function

Malwina Janiszewska, Augustyn Markiewicz, Monika Mokrzycka (2020)

Applications of Mathematics

The aim of the paper is to present a procedure for the approximation of a symmetric positive definite matrix by symmetric block partitioned matrices with structured off-diagonal blocks. The entropy loss function is chosen as approximation criterion. This procedure is applied in a simulation study of the statistical problem of covariance structure identification.

Bound for the largest singular value of nonnegative rectangular tensors

Jun He, Yan-Min Liu, Hua Ke, Jun-Kang Tian, Xiang Li (2016)

Open Mathematics

In this paper, we give a new bound for the largest singular value of nonnegative rectangular tensors when m = n, which is tighter than the bound provided by Yang and Yang in “Singular values of nonnegative rectangular tensors”, Front. Math. China, 2011, 6, 363-378.

Bounds for index of a modified graph

Bo Zhou (2004)

Discussiones Mathematicae Graph Theory

If a graph is connected then the largest eigenvalue (i.e., index) generally changes (decreases or increases) if some local modifications are performed. In this paper two types of modifications are considered: (i) for a fixed vertex, t edges incident with it are deleted, while s new edges incident with it are inserted; (ii) for two non-adjacent vertices, t edges incident with one vertex are deleted, while s new edges incident with the other vertex are inserted. ...

Bounds for sine and cosine via eigenvalue estimation

Pentti Haukkanen, Mika Mattila, Jorma K. Merikoski, Alexander Kovacec (2014)

Special Matrices

Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, [...] Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain...

Bounds for the (Laplacian) spectral radius of graphs with parameter α

Gui-Xian Tian, Ting-Zhu Huang (2012)

Czechoslovak Mathematical Journal

Let G be a simple connected graph of order n with degree sequence ( d 1 , d 2 , ... , d n ) . Denote ( α t ) i = ∑ j : i ∼ j d j α , ( α m ) i = ( α t ) i / d i α and ( α N ) i = ∑ j : i ∼ j ( α t ) j , where α is a real number. Denote by λ 1 ( G ) and μ 1 ( G ) the spectral radius of the adjacency matrix and the Laplacian matrix of G , respectively. In this paper, we present some upper and lower bounds of λ 1 ( G ) and μ 1 ( G ) in terms of ( α t ) i , ( α m ) i and ( α N ) i . Furthermore, we also characterize some extreme graphs which attain these upper bounds. These results theoretically improve and generalize some known results.

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