Antieigenvalue techniques in statistics. (English) Zbl 1168.15006

Given an operator \(T\) on a Hilbert space \(H\), the first antieigenvalue of \(T\), also called the (real) cosine of \(T\), denoted by \(\mu_1(T)\) or \(\cos T\), was first defined by K. Gustafson [Bull. Am. Math. Soc. 74, 488–492 (1968; Zbl 0172.40702)] to be
\[ \mu_1(T)= \inf_{Tf\neq 0}\frac{\text{Re}(Tf,f)}{\| Tf\|\, \| f\|}\,. \]
The author continues to explore two techniques used in antieigenvalue computations in previous papers. The first is basically a two nonzero component property for certain class of functionals. The second consists on converting the matrix optimization problems in statistics to a convex programming problem, finding the minimum of a convex function on the numerical range of an operator. The two techniques will permit generalizing some of the matrix optimization problems arising in statistics to strongly accretive operators on finite or infinite dimensional Hilbert spaces.


15A18 Eigenvalues, singular values, and eigenvectors
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
62C20 Minimax procedures in statistical decision theory
90C25 Convex programming


Zbl 0172.40702
Full Text: DOI


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