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__Problem Statement:__

Consider the matrix [3 0; 0 -2]. Find its singular value decompositions

__Problem Solution__Goal is to find A = U*S*V as below

__Step1__: Find AA', A'A. In this case they both are equal and are [9 0; 0 4];

__: Find U = eig vector (AA'). Doing so gives [1 0; 0 1];__

Step2

Step2

__Step 3__: Find S = [3 0; 0 2] (I am not showing the steps)

__Step 4__: Find V = eig vector (A'A). Doing so gives [1 0; 0 1];

__Verify__: Multiply U*S*V and it should give back A.

My problem is it gives [3 0; 0 2] which is different than A = [3 0; 0 -2].

I know that if I change V to [1 0 ; 0 -1] I will get A back. But why do my computations not show this. What am I missing?

Like I said, I did the above procedure for a lot of other numbers and I get it right. Only when I have a negative value in the matrix then it seems I am missing a -1 factor which I cannot get from my procedure.

Thanks

Asif