Lower boundary of quadratic form

Question

I have a quadratic form $x^TAx$ where $x$ is an $n \times 1$ vector and $A$ is a positive definite matrix in the sense that it has only positive eigenvalues. Am I right to say that $||x^TAx|| \ge \lambda_{min}(A) ||x||^2$ where $\lambda_{min}(A)$ is the smallest eigenvalue of $A$? If not, is there another lower boundary?

Answer 1

The bound you've given works specifically in the case that $A$ is symmetric and positive definite. For the "asymmetric, positive definite" case, we have the more general bound $$ x^TAx = x^T \left( \frac{A + A^T}{2} \right)x \geq \frac 12 \lambda_{min}(A + A^T)\|x\|^2 $$