# Linear Algebra Quadratic True False

"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.

• what happens with a 2 by 2 matrix? – Will Jagy Feb 11 '14 at 3:23

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.