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"Every quadratic form $x^TAx$ with $A$ an invertible matrix is either positive definite, negative definite, or indefinite."

Is this true or false? I am just wondering does it have to be positive, negative, or indefinite? I am thinking that if the eigenvalue is a complex number then it's technically imaginary eigenvalue so does that mean that this statement is false.

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    $\begingroup$ what happens with a 2 by 2 matrix? $\endgroup$ – Will Jagy Feb 11 '14 at 3:23
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Quadratic firms are associated to symmetric matrices, and you can easily show symmetric matrices over the reals have real eigenvalues.

As for your main question: since the matrix is invertible, it can't have any zero eigenvalues. Thus if it isn't definite one way, it must be indefinite. That is, it'll have both positive and negative values.

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Since this is a quadratic form, matrix $A$ must be symmetric. You can prove (this is a very easy fact) that any symmetric matrix has only real eigenvalues.

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