Timeline for positive hermitian elements in $M_n(\mathbb{C})$

6 events

when toggle format	what		by	license	comment
Oct 30, 2011 at 8:26	answer	added	Zack Wolske		timeline score: 1
Oct 29, 2011 at 19:48	comment	added	Jack Poulson		The set of upper (lower) triangular matrices with non-negative diagonals satisfies (i), (ii), and (iii) trivially since the eigenvalues lie on the diagonal. If we call the set of such upper triangular matrices $\mathcal U$, and the set of such lower triangular matrices $\mathcal L$, then we have a variant of (iv) which is $\mathcal U + -\mathcal U + i \mathcal U + -i \mathcal U + \mathcal L + -\mathcal L + i \mathcal L + -i \mathcal L = M_n(\mathbb C)$.
Oct 29, 2011 at 14:21	comment	added	Suvrit		ah, ok. i did not read (iv) at all :-)
Oct 29, 2011 at 13:54	comment	added	spelas		But $x^∗ax$ is also hermitian matrix if $a$ is. So $x^∗Px⊂P$, and $x^∗M_n(\mathbb{C})x=M_n(\mathbb{C})$ iff $x$ is invertible. So $x^∗Px$ either does not satisfy (iv) or equals $P$.
Oct 29, 2011 at 13:07	comment	added	Suvrit		$X^*AX \ge 0$ for all $X$ if $A \ge 0$.
Oct 29, 2011 at 12:01	history	asked	spelas	CC BY-SA 3.0