## Non positive inner products

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

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