Start with \(T={\operatorname{diag}}(1,2)\) and the two coordinate projections. Apply the invertible shear \(U\). The transformed operator is \(T^U=UTU^{-1}\), and the transformed projectors are \(Q_i=UP_iU^{-1}\).
Reading the drawing
The upper row shows T, the shear U and the conjugated operator. The lower row shows both transported projectors. The changed matrix entries are read relative to the changed packet coordinates.
Drawing labels and data 25 labels
Read the original labels as selectable text. Tables preserve grid relationships; other short labels are grouped compactly.
| Row | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(1\) | \(0\) |
| Row 2 | \(0\) | \(2\) |
| Row | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(1\) | \(1\) |
| Row 2 | \(0\) | \(1\) |
| Row | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(1\) | \(1\) |
| Row 2 | \(0\) | \(2\) |
| Row | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(1\) | \(-1\) |
| Row 2 | \(0\) | \(0\) |
| Row | Column 1 | Column 2 |
|---|---|---|
| Row 1 | \(0\) | \(1\) |
| Row 2 | \(0\) | \(1\) |
Evidence note from the source edition
The transformed operator has a nonzero entry away from the diagonal in the standard basis. Relative to its transported, oblique packets, however, both blocks between distinct packets remain zero.
Indeed,
A change in raw matrix sparsity is therefore not necessarily a change in the packet transfer graph. Conjugacy preserves the exact block identities and their zero pattern. A nonunitary shear need not preserve their Euclidean norms.
When the map is not invertible
For an intertwiner \(\Phi A=\varphi(A)\Phi\), the hull identity uses the image algebra:
Replacing \(\varphi(\AA)\) with a larger target algebra can enlarge the target hull. Seed rank need not be equal relative to that larger algebra. A surjective intertwiner gives the appropriate monotonicity inequality, not automatic equality.
Read this as
Transport the space, operators and packet maps together. The invariant is the collection of commuting algebraic relationships, rather than the appearance of the matrix.