Let H0 be an arbitrary self-adjoint n × n matrix and H(n) be an n × n (random) Wigner matrix. We show that t 7→ Tr exp(H(n)−itH0) is positive definite in the average. This partially answers a long-standing conjecture. On the basis of asymptotic freeness our result implies that t 7→ τ(exp(a − itb)) is positive definite whenever the noncommutative random variables a and b are in free relation, wi...

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‎In this paper‎, ‎some algebraic and geometrical properties of the rank$-k$ numerical hulls of normal matrices are investigated‎. ‎A characterization of normal matrices whose rank$-1$ numerical hulls are equal to their numerical range is given‎. ‎Moreover‎, ‎using the extreme points of the numerical range‎, ‎the higher rank numerical hulls of matrices of the form $A_1 oplus i A_2$‎, ‎where $A_1...

RATIONALE The phosphorus storage compound in grains, phytic acid, or myo-inositol hexakisphosphate (IP6), is important for nutrition and human health, and is reportedly the most abundant organic phosphorus compound in soils. Methods for its determination have traditionally relied on complexation with iron and precipitation, acid digestion and measurement of phosphate concentration, or 31 P NMR ...

In this paper, we revisit the greatest common right divisor (GCRD) extraction from a set of polynomial matrices , with coefficients in generic field and column dimension . We give necessary sufficient conditions for matrix to be GCRD using Smith normal form compound obtained by concatenating vertically, where also describe complete degrees freedom solution link it Hermite then an algorithm cons...

in this paper, the ultraspherical operational matrices of derivatives are constructed. based on these operational matrices, two numerical algorithms are presented and analyzed for obtaining new approximate spectral solutions of a class of linear and nonlinear lane-emden type singular initial value problems. the basic idea behind the suggested algorithms is basically built on transforming the eq...

In this paper we introduce a special form of symmetric matrices that is called central symmetric $X$-form matrix and study some properties, the inverse eigenvalue problem and inverse singular value problem for these matrices.