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Search: All articles in the CMB digital archive with keyword determinant

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1. CMB Online first

Kurdyka, Krzysztof; Paunescu, Laurentiu
 Nuij type pencils of hyperbolic polynomials Nuij's theorem states that if a polynomial $p\in \mathbb{R}[z]$ is hyperbolic (i.e. has only real roots) then $p+sp'$ is also hyperbolic for any $s\in \mathbb{R}$. We study other perturbations of hyperbolic polynomials of the form $p_a(z,s): =p(z) +\sum_{k=1}^d a_ks^kp^{(k)}(z)$. We give a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which $p_a(z,s)$ is a pencil of hyperbolic polynomials. We give also a full characterization of those $a= (a_1, \dots, a_d) \in \mathbb{R}^d$ for which the associated families $p_a(z,s)$ admit universal determinantal representations. In fact we show that all these sequences come from special symmetric Toeplitz matrices. Keywords:hyperbolic polynomial, stable polynomial, determinantal representa- tion, symmetric Toeplitz matrixCategories:15A15, 30C10, 47A56

2. CMB Online first

Reichstein, Zinovy; Vistoli, Angelo
 On the dimension of the locus of determinantal hypersurfaces The characteristic polynomial $P_A(x_0, \dots, x_r)$ of an $r$-tuple $A := (A_1, \dots, A_r)$ of $n \times n$-matrices is defined as $P_A(x_0, \dots, x_r) := \det(x_0 I + x_1 A_1 + \dots + x_r A_r) \, .$ We show that if $r \geqslant 3$ and $A := (A_1, \dots, A_r)$ is an $r$-tuple of $n \times n$-matrices in general position, then up to conjugacy, there are only finitely many $r$-tuples $A' := (A_1', \dots, A_r')$ such that $p_A = p_{A'}$. Equivalently, the locus of determinantal hypersurfaces of degree $n$ in $\mathbf{P}^r$ is irreducible of dimension $(r-1)n^2 + 1$. Keywords:determinantal hypersurface, matrix invariant, $q$-binomial coefficientCategories:14M12, 15A22, 05A10

3. CMB 2016 (vol 59 pp. 585)

Lin, Minghua
 A Determinantal Inequality Involving Partial Traces Let $\mathbf{A}$ be a density matrix in $\mathbb{M}_m\otimes \mathbb{M}_n$. Audenaert [J. Math. Phys. 48 (2007) 083507] proved an inequality for Schatten $p$-norms: $1+\|\mathbf{A}\|_p\ge \|\tr_1 \mathbf{A}\|_p+\|\tr_2 \mathbf{A}\|_p,$ where $\tr_1, \tr_2$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality $1+\det \mathbf{A}\ge \det(\tr_1 \mathbf{A})^m+\det(\tr_2 \mathbf{A})^n.$ Keywords:determinantal inequality, partial trace, block matrixCategories:47B65, 15A45, 15A60

4. CMB 2016 (vol 59 pp. 311)

Ilten, Nathan; Teitler, Zach
 Product Ranks of the $3\times 3$ Determinant and Permanent We show that the product rank of the $3 \times 3$ determinant $\det_3$ is $5$, and the product rank of the $3 \times 3$ permanent $\operatorname{perm}_3$ is $4$. As a corollary, we obtain that the tensor rank of $\det_3$ is $5$ and the tensor rank of $\operatorname{perm}_3$ is $4$. We show moreover that the border product rank of $\operatorname{perm}_n$ is larger than $n$ for any $n\geq 3$. Keywords:product rank, tensor rank, determinant, permanent, Fano schemesCategories:15A21, 15A69, 14M12, 14N15
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