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No positive definite matrix

04/04/2011

It is well known that: Let A be a symmetric matrix, if x^TAx>0, \forall x\in R^n, then A is positive definite and all this eigenvalues are positive.

Now my question is: if there exists at least a vector x\in R^n, x^TAx>0; and there exists at least a vector y\in R^n, y^TAy<0, does A has at least one positive eigenvalue and at least one negative eigenvalue?

The answer is yes: Proof by contradiction.

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