Dear all,
Could anyone help me understand the following( once and for all) which
I normally do without knowing its essential meaning:
Let x be a vector and A be a matrix.
1).x'Ax is positive definite(PD),implies all eigen values of A are
> 0. It also implies x'Ax >0 for all x not = 0.
2). x'Ax is positive semi-definite(PSD), implies all eigen values of
A are >= 0. It also implies x'Ax >=0 for all x.
3).x'Ax is negative definite(ND),implies all eigen values <0. This
also implies x'Ax < 0.
4). x'AX is negative semi-definite(NSD),implies all eigen values of A
are <=0. This also implies that x'Ax <=0.
My questions are:
a). Why is it necessary to show that a quadratic is of the form (1)-
(4)? Could you provide Statistical examples where this is
necessary?. I know that a quadratic has to be positive.
b). Is there a situation in Statistics where one has to check
definiteness of a Matrix before using it?
Always say thank you and excuse my ignorance.
Yours sincerely,
June E.Simakani(Mr)
Department of Statistics
University of Transkei
Private Bag X1,Unitra 5117
Umtata,South Africa.
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