TITLE: Moore Penrose Inverse for symmetric matrix
QUESTION [5 upvotes]: Is there any proof that a MPI of symetric matrix is symmetric matrix?
Basically I need that Moore-Penroses invers of positive semidefinite matrix is positive semidefinite.
I can show that x^T(A+)x >=0. But A+ also need to be symmetric for positive semidefinite.
Thank you for your help.

REPLY [6 votes]: Let $A$ be a symmetric matrix and let $B$ be its pseudoinverse. By definition, this means that $B$ is the unique matrix satisfying:

$ABA=A$
$BAB=B$
$AB$ and $BA$ are symmetric

Taking transposes of these equations implies that:

$AB^TA=A$
$B^TAB^T=B^T$
$B^TA$ and $AB^T$ are symmetric

Thus $B^T$ is also a pseudoinverse for $A$. By uniqueness, it follows that $B$ is symmetric.