= Similarity transformation
{title2=$A\mapsto S^{-1}AS$}
A <similarity transformation> replaces a <square matrix> by a similar <matrix>. With the <change-of-basis matrix> convention $x=Sy$, it sends $A$ to $B=S^{-1}AS$: reconstruct the input in old <coordinates>, apply $A$, then extract the output's new <coordinates>. These three operations explain the order of the factors.
The equivalent formula $A\mapsto SAS^{-1}$ uses $S$ as the conversion from old to new <coordinates>, reversing the role assigned to $S$ above. For the opening example's numerical $S$, this opposite conversion gives $SAS^{-1}=\begin{pmatrix}2&3\\0&5\end{pmatrix}$. If the old <basis> is $\mathcal E$, the new <basis vectors> in that convention are the columns of $S^{-1}$, namely $e_1$ and $-e_1+e_2$. Both formulas are <similarity transformations>; the direction of the coordinate conversion decides which one describes a specified <basis>.
An especially useful form is the <intertwining operator> equation
$$
AS=SB.
$$
It says that applying $B$ before reconstruction by $S$ gives the same physical output as reconstructing first and applying $A$. Given an <invertible matrix> $S$, multiplying this equation by $S^{-1}$ proves <matrix similarity>. Without invertibility, the equation alone does not prove <matrix similarity>.
\Image[/matrix-similarity-intertwining-invariants.png]
{title=<Matrix similarity> makes coordinate conversion commute with the action of a <linear operator>}
{description=For the worked example, both routes give the old output <coordinate vector> $(4,5)^T$. The bottom strip records <matrix similarity invariants>; the <eigenvector> tuple changes while reconstructing to the same vertical <vector>.}
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