  Semidefinite programs (SDPs) -- some of the most useful and versatile
optimization problems of the last few decades -- are often pathological: the
optimal values of the primal and dual problems may differ and may not be
attained. Such SDPs are both theoretically interesting and often impossible to
solve; yet, the pathological SDPs in the literature look strikingly similar.
  Based on our recent work \cite{Pataki:17} we characterize pathological
semidefinite systems by certain {\em excluded matrices}, which are easy to spot
in all published examples. Our main tool is a normal (canonical) form of
semidefinite systems, which makes their pathological behavior easy to verify.
The normal form is constructed in a surprisingly simple fashion, using mostly
elementary row operations inherited from Gaussian elimination. The proofs are
elementary and can be followed by a reader at the advanced undergraduate level.
  As a byproduct, we show how to transform any linear map acting on symmetric
matrices into a normal form, which allows us to quickly check whether the image
of the semidefinite cone under the map is closed. We can thus introduce readers
to a fundamental issue in convex analysis: the linear image of a closed convex
set may not be closed, and often simple conditions are available to verify the
closedness, or lack of it.
