  Generalized inverses are important in statistics and other areas of applied
matrix algebra. A \emph{generalized inverse} of a real matrix $A$ is a matrix
$H$ that satisfies the Moore-Penrose (M-P) property $AHA=A$. If $H$ also
satisfies the M-P property $HAH=H$, then it is called \emph{reflexive}.
Reflexivity of a generalized inverse is equivalent to minimum rank, a highly
desirable property. We consider aspects of symmetry related to the calculation
of various \emph{sparse} reflexive generalized inverses of $A$. As is common,
we use (vector) 1-norm minimization for both inducing sparsity and for keeping
the magnitude of entries under control.
  When $A$ is symmetric, a symmetric $H$ is highly desirable, but generally
such a restriction on $H$ will not lead to a 1-norm minimizing reflexive
generalized inverse. We investigate a block construction method to produce a
symmetric reflexive generalized inverse that is structured and has guaranteed
sparsity. Letting the rank of $A$ be $r$, we establish that the 1-norm
minimizing generalized inverse of this type is a 1-norm minimizing symmetric
generalized inverse when (i) $r=1$ and when (ii) $r=2$ and $A$ is nonnegative.
  Another aspect of symmetry that we consider relates to another M-P property:
$H$ is \emph{ah-symmetric} if $AH$ is symmetric. The ah-symmetry property is
sufficient for a generalized inverse to be used to solve the least-squares
problem $\min\{\|Ax-b\|_2:~x\in\mathbb{R}^n\}$ using $H$, via $x:=Hb$. We
investigate a column block construction method to produce an ah-symmetric
reflexive generalized inverse that is structured and has guaranteed sparsity.
We establish that the 1-norm minimizing ah-symmetric generalized inverse of
this type is a 1-norm minimizing ah-symmetric generalized inverse when (i)
$r=1$ and when (ii) $r=2$ and $A$ satisfies a technical condition.
