Complex factorisation of a psd matrix

Question

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. Assume that $A=L\cdot L^{t}$ for $L\in\mathbb{C}^{n\times m}$. Can we infer that $L$ is intact real, i.e., has only real entries?

Answer 1

Let $L=\begin{pmatrix}i&\sqrt 2\\\sqrt 2&-i\end{pmatrix}$. Then $LL'$ is the $2\times2$ identity matrix.