## View abstract

### Session S08 - Inverse Problems and Applications

Wednesday, July 14, 18:50 ~ 19:20 UTC-3

## A review on Procrustes problems for matrix inverse eigenvalue problems

### Silvia Gigola

#### Facultad de Ingeniería, Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak432bd635a32c1a6874f13fd62db90a16').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy432bd635a32c1a6874f13fd62db90a16 = 'sg&#105;g&#111;l&#97;' + '&#64;'; addy432bd635a32c1a6874f13fd62db90a16 = addy432bd635a32c1a6874f13fd62db90a16 + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_text432bd635a32c1a6874f13fd62db90a16 = 'sg&#105;g&#111;l&#97;' + '&#64;' + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloak432bd635a32c1a6874f13fd62db90a16').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy432bd635a32c1a6874f13fd62db90a16 + '\'>'+addy_text432bd635a32c1a6874f13fd62db90a16+'<\/a>';

The inverse eigenvalue problem consists of the reconstruction of a matrix from given spectral data. This kind of problems occurs in different engineering areas and arises in various applications.

Given a matrix $X$ and a diagonal matrix $D$, we are looking for solutions of the equation $AX =XD$, where $A$ is a matrix with a prescribed structure and a predefined spectrum. Based on these restrictions on matrix $A$, a variety of inverse eigenvalue problems arises.

The Procrustes problem, or the best approximation problem, associated to the inverse eigenvalue one can be described synthetically as follows: given an experimentally obtained matrix, the problem consists on finding a matrix from the problem solution set, such that it is the best approximation to the data matrix.

We will show the existence of the solutions of the inverse eigenvalue problem and the associated Procrustes problem for three kind of matrices: Hermitian reflexive matrices with respect to a normal and ${k+1}$-potent matrix, normal $J$-Hamiltonian matrices, and normal $J$-skew Hamiltonian matrices.

Joint work with Néstor Thome (Universitat Politècnica de València, Spain) and Leila Lebtahi (Universitat de València, Spain).