João Filipe Queiró

SINGULAR VALUES AND INVARIANT FACTORS OF MATRIX SUMS AND PRODUCTS

Abstract Some remarks are made concerning the range of individual singular values of sums and products of two complex square matrices, when these are allowed to vary in their orbits under unitary equivalence. An analogous question is considered for invariant factors of products of matrices over a principal ideal domain.

Point-based calibration using a parametric representation of the general imaging model

Generic imaging models can be used to represent any camera. These models are specially suited for non-central cameras for which closed-form models do not exist. Current models are discrete and define a mapping between each pixel in the image and a straight line in 3D space. Due to difficulties in the calibration procedure and model complexity these methods have not been used in practice. The focus of our work was to relax these drawbacks. In this paper we modify the general imaging model using radial basis functions to interpolate image coordinates and 3D lines allowing both an increase in resolution (due to their continuous nature) and a more compact representation. Using this new variation of the general imaging model we also develop a new linear calibration procedure. In this process it is only required to match one 3D point to each image pixel. Also it is not required the calibration of every image pixel. As a result the complexity of the procedure is significantly decreased.

On the interlacing property for singular values and eigenvalues

Abstract A new proof of the complete interlacing theorem for singular values is presented. The technique used works equally well to obtain the interlacing theorem for eigenvalues of hermitian matrices.

The determinant of the sum of two normal matrices with prescribed eigenvalues

Abstract Given complex numbers α 1 ,...,α n , β 1 ,...,β n , what can we say about the determinant of A + B , where A ( B ) is an n × n normal matrix with eigenvalues α 1 ,...,α n (β 1 ,...,β n )? Some partial answers are offered to this question.

An inverse problem for singular values and the Jacobian of the elementary symmetric functions

Abstract Given a nonnegative diagonal matrix D , how do we find a column y such that the augmented matrix [ D ∣ y ] has a prescribed set of singular values? (Conditions for existence are known.) This question is shown to lead to a system of linear equations with an interesting matrix, some of whose properties are studied.

A bound for the determinant of the sum of two normal matrices

Hadamards determinant theorem is used to obtain an upper bound for the modulus of the determinant of the sum of two normal matrices in terms of their eigenvalues. This bound is compared with another given by M. E. Miranda.

On the cartesian decomposition of a matrix

The main result of this paper is the following Let A and B be n×n hermitian matrices with eigenvalues respectively, ordered so that and let M1 be any k×k principal submatrix of . Necessary and sufficient conditions for equality are given.

Invariant Factors as Approximation Numbers

A characterization of the invariant factors of an integral matrix as approximation numbers is given. This characterization, similar to the one known for the singular values of complex matrices, is used to prove some recent results on invariant factors.

Spectral inequalities for generalized Rayleigh quotients

Abstract We show that the s-spaces of Carlson and Sa´, with a few natural axioms added, are rich enough to develop unified abstract versions of theorems leading to very general families of inequalities involving eigenvalues of sums of Hermitian matrices, singular values of products of complex matrices, and invariant factors of products of matrices over principal-ideal domains.

Sums of orbits of integral matrices

ABSTRACT For square matrices A and B over an elementary divisor domain, we study the possible invariant factors of in terms of the invariant factors of A and B. In particular, we find the exact range of in that situation.