Displaying similar documents to “Theorems of the alternative for cones and Lyapunov regularity of matrices”

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.

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...

A matrix formalism for conjugacies of higher-dimensional shifts of finite type

Michael Schraudner (2008)

Colloquium Mathematicae

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We develop a natural matrix formalism for state splittings and amalgamations of higher-dimensional subshifts of finite type which extends the common notion of strong shift equivalence of ℤ⁺-matrices. Using the decomposition theorem every topological conjugacy between two ℤ d -shifts of finite type can thus be factorized into a finite chain of matrix transformations acting on the transition matrices of the two subshifts. Our results may be used algorithmically in computer explorations on...

Complexity of computing interval matrix powers for special classes of matrices

David Hartman, Milan Hladík (2020)

Applications of Mathematics

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Computing powers of interval matrices is a computationally hard problem. Indeed, it is NP-hard even when the exponent is 3 and the matrices only have interval components in one row and one column. Motivated by this result, we consider special types of interval matrices where the interval components occupy specific positions. We show that computing the third power of matrices with only one column occupied by interval components can be solved in cubic time; so the asymptotic time complexity...