Is xTAx always non-zero for a real, symmetric, nonsingular matrix A?

[elimax]

Basic question, I think, but I'm not sure. It is a step in a demonstration, so it would be nice if it were true.

True or false? Why? If A is a real, symmetric, nonsingular matrix, then xTAx≠0 for x≠0.

 

[Truecrimson]

Since $A$ is real and symmetric, it is diagonalizable. So you may start by thinking about $x^T A x$ in a basis in which $A$ is diagonal.