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?
Asked
Active
Viewed 37 times
0
-
Note that we this does not hold one $A$ has not fall rank. For example $L=(2+I& 2-i)^t$ would yield a psd matrix, but the rank is not full. – user382144 Sep 03 '18 at 09:07
-
Do you really mean to have transpose instead of conjugate transpose in your factorization of $A$? – kimchi lover Sep 03 '18 at 12:57
-
Yes.. ätherweise it is clear... – user382144 Sep 03 '18 at 13:36
1 Answers
1
Let $L=\begin{pmatrix}i&\sqrt 2\\\sqrt 2&-i\end{pmatrix}$. Then $LL'$ is the $2\times2$ identity matrix.
kimchi lover
- 24,277