? matrix with function entries

Hi:

Suppose A is an M-by-N matrix and its entries a_i_j are functions of
independent var x.

I have several questions as follows.

(1) How to define positive definiteness of A?

(2) g(x) is a non-negative scalar function and A is non-negative definitve
(if (1) makes sense),

=> will I(x) = convolution( g, A ) remain non-negative?
(3) Will rank(I) remain same?

(4) If A(x) = U(x)*S(x)*V'(x), I(x) = UI(x)*SI(x)*VI'(x), will there be any
simple relations

between U(x) and UI(x), S(x) and SI(x), V(x) and VI(x)?

(5) More generally, what are those conservative quantities for A(x) after
convolved with g(x)?

(6) Under what conditions will A(x) = U*S(x)*V'?

Lastly, what are those good books that discuss topics similar to the above?

Thanks,
by Cheng Cosine
Jul/13/2k6 NC


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