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 $T^D(\omega)$ 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 $T^D(\omega)$ 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 $p$, with $p(0)=1$, we construct a determinantal representation $$p=\det (I - K Z),$$ where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$, $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ 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 $p$ 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 $p(0)=1$, admits a strictly contractive determinantal representation, i.e., $p=\det(I-KZ_n)$, where $n=(n_1,...,n_k)$ is a $k$-tuple of nonnegative integers, $Z_n=\bigoplus_{r=1}^k(Z^{(r)}\otimes I_{n_r})$, $Z^{(r)}=[z^{(r)}_{ij}]$ are complex matrices, $p$ is a polynomial in the matrix entries $z^{(r)}_{ij}$, and $K$ 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