Extrema of sums of heterogeneous quadratic forms
type: Item
instance of: scholarly article
title: Extrema of sums of heterogeneous quadratic forms
zbMATH Open document ID: 0893.65030
DOI: 10.1016/S0024-3795(97)00230-9
author: Marianna Bolla
author: György Michaletzky
author: Gábor Tusnády
author: Margit Ziermann
published in: Linear Algebra and its Applications
publication date: 1998-08-24T00:00:00Z
review text: This big paper focuses on the following problem arising in various situations in multivariate statistical analysis. Find the maximum of the quadratic functional \[ f_{ \text{quad}} (X)=\sum^K_{i=1} x^T_iA_ix_i \] under the constraint that the vectors \(x_1, \dots, x_k\in \mathbb{R}^n\) form an orthonormal system where \(A_1, \dots, A_k\) \((k\leq n)\) are given \(k\) symmetric positive definite \(n\times n\) matrices and the \(n\times k\) matrix \(X=[x_1, \dots, x_k]\) contains the enumerated vectors as their columns. The set of orthonormal \(k\)-tuples in \(\mathbb{R}^n\) is called a Stiefel manifold and denoted by \(V_{n,k}\); \(X\in V_{n,k}\) when the set of column vectors of the \(n\times k\) matrix \(X\) is an element of \(V_{n,k}\). As \(V_{n,k}\) is a compact manifold and \(f_{\text{quad}}\) is continuous on \(V_{n,k}\) a finite global maximum exists and it is attained at some point. To characterize the critical points of the functional let us denote by \(A(X)= [A_1x_1, \dots, A_kx_k]\) the \(n\times k\) matrix. The main result is the following theorem. Theorem 3.1. \(X\in V_{n,k}\) is a critical point of \(f_{\text{quad}}\) if and only if \(A(X)=XS\), where \(S\) is a symmetric \(k\times k\) matrix. At an arbitrary critical point the matrix \(S=X^TA(X)\) is not necessarily positive semidefinite. The next lemma shows that this is necessary at the global maximum points of \(f_{\text{quad}}\). Lemma 3.1. If \(X\in V_{n,k}\) is a global maximum of the functional \(f_{\text{quad}}\), then the corresponding matrix \(S=X^T A(X)\) is positive semidefinite. Theorem 3.1 and Lemma 3.1 together yield that at global maximum points \(A(X)=XS\), where \(S\) is a positive semidefinite matrix. Beyond it a computational iteration algorithm is proposed. Choosing an arbitrary initial orthonormal system \(X^{(0)}= [x_1^{(0)}, \dots, x_k^{(0)}]\in V_{n,k}\), the sequence \(X^{(1)}, X^{(2)}, \dots\) in \(V_{n,k}\) is constructed in the following way: from the \(m\)th element of th…
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 65K05
Mathematics Subject Classification ID: 62J05
Mathematics Subject Classification ID: 90C20
Mathematics Subject Classification ID: 65C99
zbMATH DE Number: 1100776
zbMATH Keywords: convergence
zbMATH Keywords: multivariate statistical analysis
zbMATH Keywords: maximum
zbMATH Keywords: quadratic functional
zbMATH Keywords: Stiefel manifold
zbMATH Keywords: critical points
zbMATH Keywords: iteration algorithm
zbMATH Keywords: polar decomposition
MaRDI profile type: Publication
cites work: The geometry of matrix eigenvalue methods
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full work available at URL: s0024-3795(97)00230-9
OpenAlex ID: W1965756580
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description: scientific article; zbMATH DE number 1100776