Posted by Terry J. Hilsabeck on August 02, 1998 at 01:45:31:

In Reply to: analytical eigenvalue problem posted by Patrick van der Smagt on September 03, 1997 at 06:12:05:

Why don't you just solve the eigenvalue problem

and then you will know the maximum and minimum?

: I keep hoping that someone knows some paper, book,

: or other reference which may help me further here.

: I have a matrix

: Q = sum x x + x y + y x + y y

: ij p i j i j i j i j

: where each x and y also has an index p (not shown).

: Furthermore, for all k, E{x_k} = E{y_k} = 0.

: Therefore, Q consists of two correlation matrices

: which are each other's transposes, added to two

: covariance matrices. By summing three symmetric

: pos.def. matrices, Q itself is s.p.d.

: With the courant-fischer minimax theorem I can find

: a lower bound for the eigenvalues of Q, namely:

: they are (assuming that |x_k| = 1) of order 1

: (what, however, if |x_k| <= 1, or unbounded??).

: But..... how do I find an upper bound???

: Basically I want to know if Q is better conditioned

: through the addition of the xx, xy, and yx terms.

: Second, related problem: if I take a matrix

: Q' = sum y y

: ij p i j

: where E{y_k} != 0 (summed over p), this matrix appears

: to be very badly conditioned (in my case). Does the

: condition improve when I center the y's? Does the

: condition improve by adding the terms xy, yx, and xx?