"Every quadratic form xT^ Ax with A an invertible matrix is either positive definite, negative definite, or indefinite." Is this true or false?

pominjaneh6 2022-08-12 Answered
"Every quadratic form x T A x with A an invertible matrix is either positive definite, negative definite, or indefinite."
Is this true or false?
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Answers (2)

Ashlynn Stephens
Answered 2022-08-13 Author has 25 answers
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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crazygbyo
Answered 2022-08-14 Author has 3 answers
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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