3. Change coordinates and transport an operator
All examples · Glossary: coordinate change · Papers, proofs, Lean, and code
Let \(x\) and \(y\) be two coordinate descriptions of the same object, related by \(y=Cx\). This convention fixes the direction of the change.
Change the coordinates
Take
\[C=\begin{pmatrix}1&1/2\\0&1\end{pmatrix},\qquad
x=\begin{pmatrix}2\\3\end{pmatrix}.
\]
Then
\[y=Cx=\begin{pmatrix}2+(1/2)3\\3\end{pmatrix}
=\begin{pmatrix}7/2\\3\end{pmatrix}.
\]
Undo the added half of the second coordinate with
\[C^{-1}=\begin{pmatrix}1&-1/2\\0&1\end{pmatrix}.
\]
The off-diagonal entry in \(C^{-1}C\) is \(1/2-1/2=0\), and its diagonal entries are both one. Applying the inverse to \(y\) gives
\[C^{-1}y=\begin{pmatrix}7/2-3/2\\3\end{pmatrix}
=\begin{pmatrix}2\\3\end{pmatrix}.
\]
Describe the same operation in the new coordinates
In the \(x\) coordinates, let the operation double the first coordinate:
\[D=\begin{pmatrix}2&0\\0&1\end{pmatrix}.
\]
Starting from \(y\), first recover \(x\), then apply \(D\), then return to \(y\) coordinates. The matrix is therefore \(D_y=CDC^{-1}\):
\[CD=\begin{pmatrix}2&1/2\\0&1\end{pmatrix},\qquad
D_y=\begin{pmatrix}2&1/2\\0&1\end{pmatrix}
\begin{pmatrix}1&-1/2\\0&1\end{pmatrix}
=\begin{pmatrix}2&-1/2\\0&1\end{pmatrix}.
\]
Compare the two routes
Operate first, then change coordinates:
\[Dx=\begin{pmatrix}4\\3\end{pmatrix},\qquad
CDx=\begin{pmatrix}4+3/2\\3\end{pmatrix}
=\begin{pmatrix}11/2\\3\end{pmatrix}.
\]
Change coordinates first, then use the transported operator:
\[D_yy=\begin{pmatrix}2(7/2)-(1/2)3\\3\end{pmatrix}
=\begin{pmatrix}11/2\\3\end{pmatrix}.
\]
Both routes agree. The matrix identity is \(D_yC=CD\). Changing coordinates without changing the operator matrix would generally describe a different operation.
Try it
Repeat the two routes for \(x=(1,2)^{\mathsf T}\).
Answer: \(y=(2,2)^{\mathsf T}\) and both routes produce \((3,2)^{\mathsf T}\) in the new coordinates.
Continue reading: Conjugation, commuting diagram, and operator transport.
Papers, proofs, Lean, and code
Terms in this example: Coordinates · Coordinate change · Inverse · Operator conjugation · Linear isomorphism
Use Paper I for exact packet coordinates and the library atlas for general coordinate changes.
Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates
Read · Written proofs · Lean coverage · Code and setup · Evidence
Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: coordinates(). This checks the displayed rational, modular or finite-set calculations exactly.
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