nonsingularity condition for matrix

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xueqi on 4 Aug 2013
Hi,
Assume A, B, C are 1*3 vectors and P is a 3*1 vector. For the positive definite condition of the matrix[AP,CP;CP,BP] this inequality should be satisfied
if true
% (A*P)(B*P)>(C*P)^2
end
I get the feeling that P should be canceled out and so it is unrelated to the solution but I don't know how to start...Sorry this is not a matlab question but a matrix question. I just feel that you should be familiar with matrix if you are using MatrixLab....
Thanks!
Xueqi
Roger Stafford on 6 Aug 2013
If the condition you describe,
(A*P)(B*P)>(C*P)^2,
is to hold for all possible non-zero P, it is a very strong constraint on vectors A, B, and C. It can be true only if A, B, and C are all parallel, if A and B are in the same direction, and if the product of the norms of A and B is greater than the square of the norm of C.

the cyclist on 4 Aug 2013
One way to approach to understanding this (mathematically, as you say, more MATLAB) would be to recognize that when
A = [A1 A2 A3];
and
P = [P1; P2; P3];
then
A*P = A1*P1 + A2*P2 + A3*P3
and similarly for the other terms in the equation. You can write out all of those relationships to get a feel for what this equation represents.
[I don't think P drops out.]