# Sweeping a matrix rotates its graph

Interesting!

I recently learned about a curious operation on square matrices known as sweeping, which is used in numerical linear algebra (particularly in applications to statistics), as a useful and more robust variant of the usual Gaussian elimination operations seen in undergraduate linear algebra courses. Given an \$latex {n times n}&fg=000000\$ matrix \$latex {A := (a_{ij})_{1 leq i,j leq n}}&fg=000000\$ (with, say, complex entries) and an index \$latex {1 leq k leq n}&fg=000000\$, with the entry \$latex {a_{kk}}&fg=000000\$ non-zero, the sweep \$latex {hbox{Sweep}_k[A] = (hat a_{ij})_{1 leq i,j leq n}}&fg=000000\$ of \$latex {A}&fg=000000\$ at \$latex {k}&fg=000000\$ is the matrix given by the formulae

\$latex displaystyle hat a_{ij} := a_{ij} – frac{a_{ik} a_{kj}}{a_{kk}}&fg=000000\$

\$latex displaystyle hat a_{ik} := frac{a_{ik}}{a_{kk}}&fg=000000\$

\$latex displaystyle hat a_{kj} := frac{a_{kj}}{a_{kk}}&fg=000000\$

\$latex displaystyle hat a_{kk} := frac{-1}{a_{kk}}&fg=000000\$

for all \$latex {i,j in {1,dots,n} backslash {k}}&fg=000000\$. Thus for instance if \$latex {k=1}&fg=000000\$, and \$latex {A}&fg=000000\$ is written in…

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