# Non positive inner products

1. Apr 13, 2010

### tohauz

1. Suppose v_1, v_2, . . . v_k are non-zero vectors is R^n such that (v_i,v_j)<= 0 for all i,j. Determine, with proof, the maximal possible k for n = 3, and also for arbitrary n.
2. A is 2x2, A(1,1)=A(2,2)=x-1, A(1,2)=1, A(2,1)=0. Find invertible P,Q such that
P*A*Q is diagonal. I tried singular value decomposition, but calculations are getting nasty.
Please, give me hints for these problems. Thanks a lot

2. Apr 18, 2010

### tohauz

OK, I got the first one, hopefully somebody can help me 2nd.