Displaying similar documents to “On an extension of Fekete’s lemma”

Nonsingularity, positive definiteness, and positive invertibility under fixed-point data rounding

Jiří Rohn (2007)

Applications of Mathematics

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For a real square matrix A and an integer d ≥ 0 , let A ( d ) denote the matrix formed from A by rounding off all its coefficients to d decimal places. The main problem handled in this paper is the following: assuming that A ( d ) has some property, under what additional condition(s) can we be sure that the original matrix A possesses the same property? Three properties are investigated: nonsingularity, positive definiteness, and positive invertibility. In all three cases it is shown that there exists...

On the matrix negative Pell equation

Aleksander Grytczuk, Izabela Kurzydło (2009)

Discussiones Mathematicae - General Algebra and Applications

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Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries. We give necessary and suficient conditions for solvability of the matrix negative Pell equation (P) X² - dY² = -I with d ∈ N for nonsingular X,Y belonging to M₂(Z) and his generalization (Pn) ∑ i = 1 n X ₂ i - d ∑ i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i ∈ M ₂ ( Z ) , i=1,...,n.

Nested matrices and inverse M -matrices

Jeffrey L. Stuart (2015)

Czechoslovak Mathematical Journal

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Given a sequence of real or complex numbers, we construct a sequence of nested, symmetric matrices. We determine the L U - and Q R -factorizations, the determinant and the principal minors for such a matrix. When the sequence is real, positive and strictly increasing, the matrices are strictly positive, inverse M -matrices with symmetric, irreducible, tridiagonal inverses.

The Re-nonnegative definite solutions to the matrix equation A X B = C

Qing Wen Wang, Chang Lan Yang (1998)

Commentationes Mathematicae Universitatis Carolinae

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An n × n complex matrix A is called Re-nonnegative definite (Re-nnd) if the real part of x * A x is nonnegative for every complex n -vector x . In this paper criteria for a partitioned matrix to be Re-nnd are given. A necessary and sufficient condition for the existence of and an expression for the Re-nnd solutions of the matrix equation A X B = C are presented.

Generalized induced norms

S. Hejazian, M. Mirzavaziri, Mohammad Sal Moslehian (2007)

Czechoslovak Mathematical Journal

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Let ∥ · ∥ be a norm on the algebra ℳ n of all n × n matrices over ℂ . An interesting problem in matrix theory is that “Are there two norms ∥ · ∥ 1 and ∥ · ∥ 2 on ℂ n such that ∥ A ∥ = max { ∥ A x ∥ 2 ∥ x ∥ 1 = 1 } for all A ∈ ℳ n ?” We will investigate this problem and its various aspects and will discuss some conditions under which ∥ · ∥ 1 = ∥ · ∥ 2 .

Linear preservers of row-dense matrices

Sara M. Motlaghian, Ali Armandnejad, Frank J. Hall (2016)

Czechoslovak Mathematical Journal

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Let 𝐌 m , n be the set of all m × n real matrices. A matrix A ∈ 𝐌 m , n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T : 𝐌 m , n → 𝐌 m , n that preserve or strongly preserve row-dense matrices, i.e., T ( A ) is row-dense whenever A is row-dense or T ( A ) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ 𝐌 n , m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure...