TITLE: Factorization of an invertible symmetric matrix
QUESTION [6 upvotes]: Given any invertible symmetric matrix:
$A=\begin{bmatrix}a&b&c\\ b&d&e\\ c&e&f\end{bmatrix}$
over the complex number,
Can be it factored as $A=T^\top T$?
where $T^\top$ is the transpose matrix of $T$, for some invertible matrix $T$.
Any suggestions are welcome!
Thanks!

REPLY [2 votes]: Consider the example of
$$ T = \begin{bmatrix} 
\sqrt{\left(a-\frac{c^2}{f}\right) + \frac{(c e-b f)^2}{f ( e^2-d f)} } & 0 & 0 \\ 
\frac{b f - c e}{\sqrt{f ( d f-e^2)}} & \frac{\sqrt{d f-e^2}}{\sqrt{f}} & 0 \\
\frac{c}{\sqrt{f}} & \frac{e}{\sqrt{f}} & \sqrt{f} \end{bmatrix}$$
where if you multiply it out as $T^\top T=A$. Try it!