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The 123 theorem of Probability Theory and Copositive Matrices

Alexander Kovačec, Miguel M. R. Moreira, David P. Martins (2014)

Special Matrices

Alon and Yuster give for independent identically distributed real or vector valued random variables X, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derive these using copositive matrices instead. By the same method we also give estimates for the real valued case, involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li and Li. Furthermore, we formulate a version of the above inequalities as an integral inequality...

The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland (2020)

Applications of Mathematics

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A , B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator, then one version of this quotient is the spectral radius of A . In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with...

The Frisch scheme in algebraic and dynamic identification problems

Roberto P. Guidorzi, Roberto Diversi, Umberto Soverini (2008)

Kybernetika

This paper considers the problem of determining linear relations from data affected by additive noise in the context of the Frisch scheme. The loci of solutions of the Frisch scheme and their properties are first described in the algebraic case. In this context two main problems are analyzed: the evaluation of the maximal number of linear relations compatible with data affected by errors and the determination of the linear relation actually linking the noiseless data. Subsequently the extension...

The higher rank numerical range of nonnegative matrices

Aikaterini Aretaki, Ioannis Maroulas (2013)

Open Mathematics

In this article the rank-k numerical range ∧k (A) of an entrywise nonnegative matrix A is investigated. Extending the notion of elements of maximum modulus in ∧k (A), we examine their location on the complex plane. Further, an application of this theory to ∧k (L(λ)) of a Perron polynomial L(λ) is elaborated via its companion matrix C L.

The minimum, diagonal element of a positive matrix

M. Smyth, T. West (1998)

Studia Mathematica

Properties of the minimum diagonal element of a positive matrix are exploited to obtain new bounds on the eigenvalues thus exhibiting a spectral bias along the positive real axis familiar in Perron-Frobenius theory.

The P 0 -matrix completion problem.

Choi, Ji Young, DeAlba, Luz Maria, Hogben, Leslie, Maxwell, Mandi S., Wangsness, Amy (2002)

ELA. The Electronic Journal of Linear Algebra [electronic only]

The theory and applications of complex matrix scalings

Rajesh Pereira, Joanna Boneng (2014)

Special Matrices

We generalize the theory of positive diagonal scalings of real positive definite matrices to complex diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite matrix M if there exists an invertible complex diagonal matrix D such that A = D*MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive definite matrix has at most 2n−1 scalings...

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