Differential and geometric properties of Rayleigh quotients with applications

the following cost functions: In this paper, learning rules are proposed for simultaneous corn- GI(U) = tr{(UTU)(UTBU)-lD, (la) putation of minor eigenvectors of a covariance matrix. To understand the optimality conditions of Rayleigh quotients, many Fi (U) = tr (UTBU) (UT U) -l D. (lb) interesting identities and properties related are derived. For ex- t ample, it is shown that the Hessian matrix is singular at each The objective iS to maximize these functions over all full rank critical point of the Rayleigh quotient. Based on these proper- matrices U E lRnxp. Here tr(X) denotes the trace of a square ties, MCA rules are derived by optimizing a weighted inverse matrix X, (.)T denotes matrix transpose, D is a diagonal matrix Rayleigh quotient so that the optimum weights at equilibrium of size p having distinct positive eigenvalues. It will be assumed points are exactly the desired eigenvectors of a covariance ma- that D = diag(di,... dp) and that dl > d2 > ... > dp > 0. trix...

• Type: Conference
• Year: 2006
• Where: ISCAS
• Authors: M. A. Hasan
• Keywords: ISCAS 2006, Matrix, Matrix Rayleigh Quotient, Rayleigh Quotient