Prove that the quadratic form $x^\top A x$ is equal to tr($B^Txx^TB$) when $A=BB^T$

Question

Consider $x^\top A x$, where $A \in \mathbb{R}^{n \times n}$ and $x \in \mathbb{R}^n$.

Suppose $A$ can be factors into $A = BB^T$

Then claim:

$x^T Ax = x^T BB^T x = \text{trace}(B^T xx^TB)$

It is not immediately obvious to me why this is true.

It seems to be using the following equality, $\text{trace}(ab) = \text{trace}(ba)$

Does anyone have a good way to see why it works?

Answer 1

Hint: Let $y=B^T x$ and use $\mathrm{trace}(y^T y) = \mathrm{trace}(y y^T)$.