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Register a new userDynamics of products of nonnegative matrices
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The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length. We also make use of an appropriate definition of the exponential map and the logarithm map on the positive orthant of $mathbb{R}^{n}$ and explore the relationship between the periodic points of certain subhomogeneous maps defined through the above functions and the periodic points of matrix products, mentioned above.
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We examine iteration of certain skew-products on the bidisk whose components are rational inner functions, with emphasis on simple maps of the form $Phi(z_1,z_2) = (phi(z_1,z_2), z_2)$. If $phi$ has degree $1$ in the first variable, the dynamics on each horizontal fiber can be described in terms of Mobius transformations but the global dynamics on the $2$-torus exhibit some complexity, encoded in terms of certain $mathbb{T}^2$-symmetric polynomials. We describe the dynamical behavior of such mappings $Phi$ and give criteria for different configurations of fixed point curves and rotation belts in terms of zeros of a related one-variable polynomial.
The main of this work is to use the unit lower triangular matrices for solving inverse eigenvalue problem of nonnegative matrices and present the easier method to solve this problem.
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We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree $O(d)$ on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial optimization problem requires solving a semidefinite program (SDP) of size $n^{O(d)}$, with multiplicative approximation guarantees of $Omega(frac{1}{n})$. We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in $n$ and $d$, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as $d rightarrow infty$. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.
Tubal scalars are usual vectors, and tubal matrices are matrices with every element being a tubal scalar. Such a matrix is often recognized as a third-order tensor. The product between tubal scalars, tubal vectors, and tubal matrices can be done by the powerful t-product. In this paper, we define nonnegative/positive/strongly positive tubal scalars/vectors/matrices, and establish several properties that are analogous to their matrix counterparts. In particular, we introduce the irreducible tubal matrix, and provide two equivalent characterizations. Then, the celebrated Perron-Frobenius theorem is established on the nonnegative irreducible tubal matrices. We show that some conclusions of the PF theorem for nonnegative irreducible matrices can be generalized to the tubal matrix setting, while some are not. One reason is the defined positivity here has a different meaning to its usual sense. For those conclusions that can not be extended, weaker conclusions are proved. We also show that, if the nonnegative irreducible tubal matrix contains a strongly positive tubal scalar, then most conclusions of the matrix PF theorem hold.
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