Sequence Of Singular Values Research Articles

Superoptimal singular values and indices of matrix functions

LetG be a matrix-valued function on the unit circle which is the sum of a continuous function and anH ∞ function. We establish an inequality between corresponding terms of the sequence of singular values of the Hankel operator {s j (H G )} and a sequence formed from the superoptimal singular values ofG with repetitions. The number of times each superoptimal singular value is repeated is a positive integer index which is the winding number of a related scalar function on the circle and gives information about the superoptimal error functionG-Q withQ∈H ∞ . In the second part of the paper we establish a property of invariance of the sum of the indices corresponding to a particular superoptimal singular value. This establishes the truth of two conjectures made in [PeY].

Faces of the unit ball of a unitarily invariant norm

Let ψ be a unitarily invariant norm on the space of all n× m complex matrices, and let g: R n → R be the symmetric gauge function associated with ψ. That is to say, we have ψ( A) g( s( A)) for any A, where s(A) is the nonincreasing sequence of singular values of A. In this article we consider the relationship between the facial structures of the closed unit balls of g and ψ, which we denote by B g and B ψ. Our main result gives a complete characterization of the faces of B ψ in terms of the faces of B g . For, with each face E of B ψ we associate a standard face F E of B g (i.e., F E is a face of B g whose barycenter is a nonnegative, nonincreasing n-vector). Conversely, each standard face F of B g originates a set of n× m matrices, say the set M F , which is proven to be a face of B ψ. Our main result also asserts that any face E of B ψ is of the form Q M F R, where Q and R are unitary matrices of appropriate orders and F F E . We then explore the main result and draw some of its consequences. For example, we show that the singular values of the barycenter of a face E of B ψ have the same multiplicities as the barycenter of F E . Formulas are given which relate the dimensions of E and F E . The relative interior and the relative boundary of E are characterized in terms of the relative interior and the relative boundary of F E . We completely describe the face of B ψ generated by a matrix A. Our results also cover the case of real matrices.

Exposed faces and duality for symmetric and unitarily invariant norms

Let ψ be a unitarily invariant norm on the space of (real or complex) n× m matrices, and g the associated symmetric gauge function: thus ψ( A) g( s( A)), where s(A) is the decreasing sequence of singular values of A. Denote by B ψ and B g the closed unit balls of ψ and g. In a previous paper we showed a close relationship between the faces of B ψ and those of B g . In particular, to each face of B ψ we associated a standard face of B g , and we used this association to completely describe the matrices that are members of an individual face of ψ. In the present paper, we consider the duality operator that transforms each exposed face of B ψ into an exposed face of the unit ball of ψ's dual and the duality operator that does the same with exposed faces of B g . We show that these two operators are very nicely related. Among other results, we prove the following. Given an exposed face E of B ψ, the standard face associated with the dual of E is precisely the dual of the standard face of B g associated with E . E is an exposed face of B ψ if and only if its associated standard face is an exposed face of B g . As a by-product we completely determine the subdifferential of ψ in terms of the subdifferential of g, and we completely characterize the matrices that are dual to a given matrix A with respect to ψ.

Order Selection for Arx-Models Via Hankel Approximation of High Order Estimates

The identification scheme via a combination of high order least squares estimation followed by Hankel model reduction is an attractive alternative to the direct estimation schemes that have to be used when a realistic disturbance structure is assumed in practical system estimation problems. Order selection for a high order estimated model however should not be based on the residual variance since it does not, give reliable information about the true system order. It is much more informative to inspect the sequence of Hankel singular values of the high order model. This sequence is closely related to the distribution of system gain over the model order. In this paper the sequence of the Hankel singular values of high order estimates for actually low order systems is investigated by statistical simulation. It is shown to be understandable as a perturbation of the singular value sequence of the true model by a typical pattern related to a noise model. As a null hypothesis for a corresponding model order test the behavior of the singular values of such noise models fitted to two independent white noise series is investigated. An order selection procedure is developed from these observations.

A note on a system theoretic approach to a conjecture by Peller-Khrushchev

Based on the construction of infinite dimensional balanced realizations an alternative solution to the following inverse spectral problem is presented: Given a monotonically decreasing sequence of positive numbers (σ n ) n ⩾ 1 , does there exist a Hankel operator whose sequence of singular values is (σ n ) n ⩾ 1 ?