Sweeping a matrix rotates its graph


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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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