= Matrix similarity
{title2=$B=S^{-1}AS$}
{wiki}
= Similar matrix
{synonym}
= Similar matrices
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= Similarity
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Two <square matrices> $A,B$ over the same <field> are similar if $B=S^{-1}AS$ for an <invertible matrix> $S$. They can describe one <linear operator> using two different <ordered bases>, with the same <basis> used for input and output in each description. Fix the convention $x=Sy$: $x$ is the old <coordinate vector>, $y$ the new <coordinate vector>, and the columns of $S$ are the new <basis vectors> in old <coordinates>. This is the convention of the <change-of-basis matrix>.<Linear Algebra Done Right by Sheldon Axler (4th edition)>[§3D, Theorem 3.84, printed p. 93: change-of-basis formula; the matrix there converts in the opposite direction to our $S$]{ref}
For a concrete <linear operator> on the physical plane, let $T$ double horizontal components and multiply vertical components by five. In the old <standard basis> $\mathcal E=(e_1,e_2)$, its <matrix of a linear map> is $A=\operatorname{diag}(2,5)$. Choose the new <ordered basis> $\mathcal F=(b_1,b_2)$ with $b_1=e_1$, $b_2=e_1+e_2$. Then
$$
S=\begin{pmatrix}1&1\\0&1\end{pmatrix},\qquad
S^{-1}=\begin{pmatrix}1&-1\\0&1\end{pmatrix},\qquad
B=S^{-1}AS=\begin{pmatrix}2&-3\\0&5\end{pmatrix}.
$$
The second column of $B$ says $T(b_2)=-3b_1+5b_2=2e_1+5e_2$. Its off-diagonal entry records how the chosen <basis vectors> describe the output; the physical <linear operator> still stretches along the old horizontal and vertical directions.
Take the physical <vector> $p=2e_1+e_2=b_1+b_2$. Its two <coordinate vectors>, and the two descriptions of $T(p)$, are
$$
x=\begin{pmatrix}2\\1\end{pmatrix},\qquad y=\begin{pmatrix}1\\1\end{pmatrix},\qquad
Ax=\begin{pmatrix}4\\5\end{pmatrix},\qquad By=\begin{pmatrix}-1\\5\end{pmatrix}.
$$
Reconstruction gives $T(p)=4e_1+5e_2=-b_1+5b_2$. The output <coordinate vectors> differ, but the endpoint in the physical plane is the same.
\Image[/matrix-similarity-physical-bases.png]
{title=One <linear operator> and the same two physical <vectors> described in a <standard basis> and an oblique <basis>}
{description=Both panels use the same physical plane and scale. The blue arrow is $p$, and the orange arrow is $T(p)$. Dotted paths reconstruct them from their respective <basis vectors>; the right grid follows $b_1=e_1$, $b_2=e_1+e_2$.}
This is why <matrix similarity> is useful: a convenient <basis> can simplify calculations without changing the <linear operator>. A <diagonalizable matrix> can be replaced by a <diagonal matrix>, making repeated application a calculation with powers of its <eigenvalues>. In the example, $A$ makes the two independent stretch factors visible, while $B$ expresses the same action in directions chosen for another purpose. <Matrix similarity invariants> separate properties of the <linear operator> from features of its address system.
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