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Facial structures of separable and PPT states

Seung-Hyeok Kye (2011)

Banach Center Publications

A positive semi-definite block matrix (a state if it is normalized) is said to be separable if it is the sum of simple tensors of positive semi-definite matrices. A state is said to be entangled if it is not separable. It is very difficult to detect the border between separable and entangled states. The PPT (positive partial transpose) criterion tells us that the partial transpose of a separable state is again positive semi-definite, as was observed by M. D. Choi in 1982 from...

Factorizable matrices

Miroslav Fiedler, Frank J. Hall (2013)

Special Matrices

We study square matrices which are products of simpler factors with the property that any ordering of the factors yields a matrix cospectral with the given matrix. The results generalize those obtained previously by the authors.

Factorization of CP-rank- 3 completely positive matrices

Jan Brandts, Michal Křížek (2016)

Czechoslovak Mathematical Journal

A symmetric positive semi-definite matrix A is called completely positive if there exists a matrix B with nonnegative entries such that A = B B ⊤ . If B is such a matrix with a minimal number p of columns, then p is called the cp-rank of A . In this paper we develop a finite and exact algorithm to factorize any matrix A of cp-rank 3 . Failure of this algorithm implies that A does not have cp-rank 3 . Our motivation stems from the question if there exist three nonnegative polynomials of degree at most four that...

Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong (2003)

Colloquium Mathematicae

Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix ( f ( x i , x i ) ) having f evaluated at the greatest common divisor ( x i , x i ) of x i and x i as its...

Factorizations for q-Pascal matrices of two variables

Thomas Ernst (2015)

Special Matrices

In this second article on q-Pascal matrices, we show how the previous factorizations by the summation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows [...] We also find two different matrix products for [...]

Falseness of the finiteness property of the spectral subradius

Adam Czornik, Piotr Jurgas (2007)

International Journal of Applied Mathematics and Computer Science

We prove that there exist infinitely may values of the real parameter α for which the exact value of the spectral subradius of the set of two matrices (one matrix with ones above and on the diagonal and zeros elsewhere, and one matrix with α below and on the diagonal and zeros elsewhere, both matrices having two rows and two columns) cannot be calculated in a finite number of steps. Our proof uses only elementary facts from the theory of formal languages and from linear algebra, but it is not constructive...

Fault tolerant control for uncertain time-delay systems based on sliding mode control

Jun Sheng Wu, Zhengxin Weng, Zuo Hua Tian, Song Jiao Shi (2008)

Kybernetika

Fault tolerant control for uncertain systems with time varying state-delay is studied in this paper. Based on sliding mode controller design, a fault tolerant control method is proposed. By means of the feasibility of some linear matrix inequalities (LMIs), delay dependent sufficient condition is derived for the existence of a linear sliding surface which guarantees quadratic stability of the reduced-order equivalent system restricted to the sliding surface. A reaching motion controller, which can...

Fermat's Equation in Matrices

Khazanov, Alex (1995)

Serdica Mathematical Journal

The Fermat equation is solved in integral two by two matrices of determinant one as well as in finite order integral three by three matrices.

Fiedler vectors with unbalanced sign patterns

Sooyeong Kim, Stephen J. Kirkland (2021)

Czechoslovak Mathematical Journal

In spectral bisection, a Fielder vector is used for partitioning a graph into two connected subgraphs according to its sign pattern. We investigate graphs having Fiedler vectors with unbalanced sign patterns such that a partition can result in two connected subgraphs that are distinctly different in size. We present a characterization of graphs having a Fiedler vector with exactly one negative component, and discuss some classes of such graphs. We also establish an analogous result for regular graphs...

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