The following lemma ensures that the elementary Gaussian operations indeed do not change the solutions of an SLE:

Lemma: Elementary Gaussian Operations Do Not Change the Solutions of an SLE

Let an SLE be given in the form of an extended coefficient matrix $A|\gamma$ and let $S$ be the set of its solutions. Then $S$ is an invariant of the elementary Gaussian operations. In other words, if we apply any finite series of these operations $o_1,o_2,\ldots,o_k$ to $A|\gamma$, leading to the new SLEs

• SLE$_1:=A^{(1)}|\gamma^{(1)}:=o_1(A|\gamma)$ after the first opertion $o_1,$
• SLE$_2:=A^{(2)}|\gamma^{(2)}:=o_2(A^{(1)}|\gamma^{(1)})$ after the second opertion $o_2,$
• SLE$_3:=A^{(3)}|\gamma^{(3)}:=o_3(A^{2)}|\gamma^{(2)})$ after the third opertion $o_3,$
• ...
• SLE$_k:=A^{(k)}|\gamma^{(k)}:=o_k(A^{(k-1)}|\gamma^{(k-1)})$ after the $k$-th opertion $o_k,$

then the solutions $S$ of SLE$_k$ (if any), will be exactly the same as the solutions as the original SLE.

Proofs: 1

Definitions: 1

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References

Bibliography

1. Knabner, P; Barth, W.: "Lineare Algebra - Grundlagen und Anwendungen", Springer Spektrum, 2013