FIRST JOINT MATH-STAT COLLOQUIUM
by
C. R. Rao
Eberly Professor of Statistics
The Pennsylvania State University
on
Anti-eigen values and anti-singular values of a
matrix and applications to problems in statistics

 
Opening Remarks: CLAS Dean Neil Sullivan

Date: Monday, September 22, 2003
Time: 4:00 p.m.
Room: LIT 109

Refreshments: After colloquium at 5:00pm in LIT 339

 

  RAO PIC

Abstract: Let $ A$ be $ p\times p$ positive definite matrix. A $ p$-vector $ x$ such that $ Ax=\lambda x$ is called an eigenvector with the associated with eigenvalue $ \lambda$. Equivalent characterizations are: (i) $ \cos \theta=1$, where $ \theta$ is the angle between $ x$ and $ Ax$; (ii) $ (x^\prime Ax)^{-1}=xA^{-1}x$; (iii) $ \cos \Phi=1$, where $ \phi$ is the angle between $ A^{1/2}x$ and $ A^{-1/2}x$.

We ask the question what is $ x$ such that $ \cos\theta$ as defined in (i) is a minimum or the angle of separation between $ x$ and $ Ax$ is a maximum. Such a vector is called an anti-eigenvector and $ \cos\theta$ an anti-eigenvalue of $ A$. This is the basis of operator trigonometry developed by K. Gustafson and P.D.K.M. Rao (1997), Numerical Range: The Field of Values of Linear Operators and Matrices, Springer. We may define a measure of departure from condition (ii) as $ \min[(x^\prime Ax)(x^\prime A^{-1}x)]^{-1}$ which gives the same anti-eigenvalue. The same result holds if the maximum of the angle $ \Phi$ between $ A^{1/2}x$ and $ A^{-1/2}x$ as in condition (iii) is sought. We define a hierarchical series of anti-eigenvalues, and also consider optimization problems associated with measures of separation between an $ r(<p)$ dimensional subspace $ S$ and its transform $ AS$.

Similar problems are considered for a general matrix $ A$ and its singular values leading to anti-singular values.

Other possible definitions of anti-eigen and anti-singular values, and applications to problems in statistics will be presented.


Professor C.R. Rao has been a world leader in statistical science for the past six decades. He served as Director of the Indian Statistical Institute (ISI) for many years and made it into a world class center. After retirement from ISI, he moved to the USA, first serving as University Professor in Pittsburgh, and now as Eberly Professor at Penn. State University. Honors and recognitions include the Wilks Medal of the American Statistical Association, the Guy Silver Medal of the Royal Statistical Society, the Mahalanobis Gold Medal of the Indian Science Congress, and the Ramanujan Medal of the Indian National Science Academy. He has been elected as fellow of the Royal Society (FRS) as well as Member of the National Academy of Sciences, USA. Last year he was awarded the National Medal of Science by President Bush.