nonsingularity condition for matrix

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]

Xueqi, could you please explain in much greater detail what you are asking? Based on what you have said, the matrix [A,C;C,B] is not square but rather 2 x 6, and the term 'nonsingular' applies only to square matrices as far as I am aware. What exactly is your concept of nonsingularity in this case?

[xueqi]

Hi, sorry. The matrix is [AP,CP;CP,BP]. So it is a 2*2 matrix.

[Roger Stafford]

Your inequality is trivially equivalent to stating that the determinant of the (revised) 2x2 matrix is positive. That is a stronger condition than merely being nonsingular.

You need to seriously rethink the question you are posing here in my opinion.

[xueqi]

yes thanks. I was confused with non sigularity condition with postive definite condtion.

[Roger Stafford]

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.

Answer 1

[the cyclist]

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.]