Re: ? dual spaces


"Jack Schmidt" <Jack.Schmidt.SciMath@xxxxxxxxx> wrote in message
news:7577067.1184781337314.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxxxxx
I don't think there is much hope in general, but for some
specific types of nonlinear functions you may get what
you want.

For instance, work with the nonlinear function
f(x) = xAx^t, where A is an n x n matrix, x is a
1 x n row vector, and f(x) is a 1 x 1 scalar. You
can replace A by (A+A^t)/2 without changing the
function, and this matrix is symmetric, and so normal.
Write (A+A^t)/2 = UDU^t with U unitary and D diagonal,
then f(xU^t) = Sum( di xi^2, i=1..n). In some sense
then, you might be interested in those i with di=0,
thinking of it like the kernel.


This reminds me something. If we define f(x) = x'*A*x/x'*x,

where A is Hermitian and x is N-by-1. Then lambda_min <=

f(x) <= lambda_max, where lambda denotes eigenvalue of A.

Can one say that any x in A's domain can be expressed by A's

e-vectors or one cannot because this is a nonlinear map?

But one can still form a smaller subset by constraining x_con

to be the subset "spanned" (if one may write that...) by some of

the e-vectors of A. More specifically, let u_n denote e-vectors

corresponding to e-values sorted from largest to smallest e-value

lambda_n. If x_con ={u_3, u_5} and dim(A) = 7, will the image

be limited to lambda_5 <= f(x_con) <= lambda_3?




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