Large number from matrix pre and post multiplication

Question

Let us assume that $M\in\Re^{n\times n}$ is a constant matrix and $x\in\Re^{n}$ is a vector, i can choose the entries of this vector.

There is a condition such that the product:

$x^{T}Mx$

leads to the largest scalar number possible?

Answer 1

The product $x^TMx$ is a quadratic form. You can prove that $x^TMx = \frac{1}{2}x^T(M+M^T)x$ and thus it is sufficient to consider the product $x^TMx$ with $M$ symmetric. Depending on the definiteness of $M$ the quadratic form may not have a maximum for $x\in\mathbb{R}^n$. There is only a maximum of $x = 0$ when $M$ is negative-definite. It is instead usual to consider constrained optimisation of $x^TMx$, e.g. by restricting $x$ to the unit hypercube.