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Same Map, New Outfit: Similarity and Conjugation

Change the basis and an operator's matrix changes by conjugation. The matrices that arise this way are called similar, and similarity is the exact relation 'represents the same operator.' This is why we hunt for the simplest matrix in each similarity class.

How a matrix moves when the basis moves

Fix an operator T: V -> V and two bases of V. Let A be the matrix of T in the first basis, A' in the second, and let S be the change of basis matrix between them. The two matrices are related by A' = S^-1 A S. Reading right to left: translate new coordinates to old (S), apply T in old coordinates (A), translate the answer back to new (S^-1).

Two square matrices A and A' are called similar when A' = S^-1 A S for some invertible S. The whole content of similarity is this: A and A' are similar precisely when they represent the same operator in different bases. Similarity is the matrix-level shadow of 'it's the same map underneath.'

A = [2, 0;       S = [1, 1;        S^-1 = [ 1, -1;
     0, 3]            0, 1]                 0,  1]

A' = S^-1 A S
   = [1,-1; 0,1] * [2,0; 0,3] * [1,1; 0,1]
   = [1,-1; 0,1] * [2,2; 0,3]
   = [2, -1;
      0,  3]

A and A' are similar: different numbers, SAME operator.
Shared fingerprints: trace 2+3 = 5, determinant 2*3 = 6.
Conjugation by S turns A into a similar matrix A' with the same trace and determinant.

Invariants: what survives every disguise

The operation A -> S^-1 A S is called conjugation of operators. Because similar matrices are the same operator in disguise, any quantity that does NOT change under conjugation is a genuine property of the operator. These are the similarity invariants — and you already know several.

  1. Trace is invariant: tr(S^-1 A S) = tr(A), because trace ignores conjugation (tr(XY) = tr(YX)).
  2. Determinant is invariant: det(S^-1 A S) = det(S)^-1 det(A) det(S) = det(A).
  3. The characteristic polynomial is invariant, and with it the entire spectrum of eigenvalues.
  4. Rank and nullity are invariant — conjugation cannot change how much the operator collapses.

Conjugation as moving the viewpoint

There is a slogan worth memorizing: conjugation is a change of viewpoint, not a change of object. S^-1 A S means 'do A, but seen from the coordinate system S sets up.' This reframes a search for the best matrix as a search for the best basis — diagonalize when you can, and when an operator cannot be diagonalized, find the cleanest near-diagonal form instead.

This viewpoint also explains a powerful move: if a subspace U is invariant under T (meaning T(U) ⊆ U), then choosing a basis adapted to U makes the matrix block-triangular. The top-left block is the restriction of T to U, and the bottom-right block is the induced map on the quotient V/U. Similarity lets us choose the basis that exposes such structure.