221 research outputs found
A matrix and its inverse: revisting minimal rank completions
We revisit a formula that connects the minimal ranks of triangular parts of a
matrix and its inverse and relate the result to structured rank matrices. We
also address the generic minimal rank problem.Comment: 9 page
The Operator Valued Autoregressive Filter Problem and the Suboptimal Nehari Problem in Two Variables
Necessary and sufficient conditions are given for the solvability of the
operator valued two-variable autoregressive filter problem. In addition, in the
two variable suboptimal Nehari problem sufficient conditions are given for when
a strictly contractive little Hankel has a strictly contractive symbol.Comment: 18 page
A Linear-algebraic Proof of Hilbert's Ternary Quartic Theorem
Hilbert's ternary quartic theorem states that every nonnegative degree 4
homogeneous polynomial in three variables can be written as a sum of three
squares of homogeneous quadratic polynomials. We give a linear-algebraic
approach to Hilbert's theorem by showing that a structured cone of positive
semidefinite matrices is generated by rank 1 elements
The Normal Defect of Some Classes of Matrices
An n \times n matrix A has a normal defect of k if there exists an (n+k)
\times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and
k minimal. In this paper we compute the normal defect of a special class of 4
\times 4 matrices, namely matrices whose only nonzero entries lie on the
superdiagonal, and we provide details for constructing minimal normal
completion matrices A_{ext}. We also prove a result for a related class of n
\times n matrices. Finally, we present an example of a 6 \times 6 block
diagonal matrix having the property that its normal defect is strictly less
than the sum of the normal defects of each of its blocks, and we provide
sufficient conditions for when the normal defect of a block diagonal matrix is
equal to the sum of the normal defects of each of its blocks.Comment: 17 page
Outer factorizations in one and several variables
A multivariate version of Rosenblum's Fejer-Riesz theorem on outer
factorization of trigonometric polynomials with operator coefficients is
considered. Due to a simplification of the proof of the single variable case,
new necessary and sufficient conditions for the multivariable outer
factorization problem are formulated and proved.Comment: 19 page
Fractional Minimal Rank
The notion of fractional minimal rank of a partial matrix is introduced, a
quantity that lies between the triangular minimal rank and the minimal rank of
a partial matrix. The fractional minimal rank of partial matrices whose
bipartite graph is a minimal cycle are determined. Along the way, we determine
the minimal rank of a partial block matrix with invertible given entries that
lie on a minimal cycle. Some open questions are stated.Comment: 18 pages, no figure
On the augmented Biot-JKD equations with Pole-Residue representation of the dynamic tortuosity
In this paper, we derive the augmented Biot-JKD equations, where the memory
terms in the original Biot-JKD equations are dealt with by introducing
auxiliary dependent variables. The evolution in time of these new variables are
governed by ordinary differential equations whose coefficients can be
rigorously computed from the JKD dynamic tortuosity function by
utilizing its Stieltjes function representation derived in
\cite{ou2014on-reconstructi}, where an algorithm for computing the pole-residue
representation of the JKD tortuosity is also proposed. The two numerical
schemes presented in the current work for computing the poles and residues
representation of improve the previous scheme in the sense that
they interpolate the function at infinite frequency and have much higher
accuracy than the one proposed in \cite{ou2014on-reconstructi}
Norm-constrained determinantal representations of polynomials
For every multivariable polynomial , with , we construct a
determinantal representation where is a diagonal
matrix with coordinate variables on the diagonal and is a complex square
matrix. Such a representation is equivalent to the existence of whose
principal minors satisfy certain linear relations. When norm constraints on
are imposed, we give connections to the multivariable von Neumann inequality,
Agler denominators, and stability. We show that if a multivariable polynomial
, satisfies the von Neumann inequality, then admits a
determinantal representation with a contraction. On the other hand, every
determinantal representation with a contractive gives rise to a rational
inner function in the Schur--Agler class
Contractive determinantal representations of stable polynomials on a matrix polyball
We show that an irreducible polynomial with no zeros on the closure of a
matrix unit polyball, a.k.a. a cartesian product of Cartan domains of type I,
and such that , admits a strictly contractive determinantal
representation, i.e., , where is a -tuple
of nonnegative integers, ,
are complex matrices, is a polynomial in the
matrix entries , and is a strictly contractive matrix. This
result is obtained via a noncommutative lifting and a theorem on the
singularities of minimal noncommutative structured system realizations
Rational inner functions on a square-matrix polyball
We establish the existence of a finite-dimensional unitary realization for
every matrix-valued rational inner function from the Schur--Agler class on a
unit square-matrix polyball. In the scalar-valued case, we characterize the
denominators of these functions. We also show that every polynomial with no
zeros in the closed domain is such a denominator. One of our tools is the
Kor\'{a}nyi--Vagi theorem generalizing Rudin's description of rational inner
functions to the case of bounded symmetric domains; we provide a short
elementary proof of this theorem suitable in our setting
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