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Change of Basis and Similar Matrices

The same geometric transformation looks like completely different matrices depending on which coordinate system you describe it in. Changing basis with B^(-1)*A*B is just translating one observer's description into another's — and matrices related this way, called similar, secretly share their most important numbers.

One map, many descriptions

A transformation is a geometric fact — it moves space the same way no matter who watches. But the *matrix* that describes it depends on your basis, because a matrix records what happens to the basis vectors. Swap to a different basis and the same map gets a different coordinate description, hence a different matrix.

Translating with B^(-1)*A*B

Let B hold the new basis vectors as its columns. To apply the map to a vector given in new coordinates, sandwich it: B sends new coordinates to standard ones, A does the transformation there, and B^(-1) brings the result back. The change-of-basis formula reads the same map in the new system.

  1. Read it right to left: B^(-1) ( A ( B v ) ) — translate in, transform, translate out.
  2. The result A' = B^(-1)*A*B is the matrix of the SAME transformation, seen in the new basis.
  3. Diagonalization is exactly this with B = the eigenvectors: B^(-1)*A*B = D, the simplest possible view.

Similar matrices share their fingerprints

Two matrices related by A' = B^(-1)*A*B are called similar. Since they describe the same underlying map, they must agree on every quantity that belongs to the map itself, not to the coordinates. The three you should remember: trace, determinant, and the eigenvalues.

A = [[2,1],[0,3]]   trace = 2+3 = 5,  det = 6,  eigenvalues 2,3
D = [[2,0],[0,3]]   trace = 2+3 = 5,  det = 6,  eigenvalues 2,3
same map, same fingerprints
A and its diagonal form are similar, so trace, determinant, and eigenvalues all match.