Similar Matrices and Change of Basis

Two matrices are similar exactly when they describe the same linear transformation in different coordinate systems. This is the right notion of equivalence for square matrices: the matrix changes when you change the basis, but the underlying transformation does not. Whatever the linear map "really does" geometrically, how much it rotates, stretches, projects, survives the change of basis. The numerical features that survive are the similarity invariants, and they are exactly the features worth computing.

The definition

Two $n \times n$ matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ with $B = P^{-1} A P$. The matrix $P$ is the change-of-basis matrix: its columns are the new basis vectors expressed in the old coordinates. If you apply $A$ to a vector and then re-express the result in a new basis, you get the same answer as if you had first re-expressed the input in the new basis, applied $B$, and stayed there.

Similarity invariants

The following quantities are preserved by similarity:

• Determinant: $\det(B) = \det(P^{-1} A P) = \det(P)^{-1} \det(A) \det(P) = \det(A)$.
• Trace: $\operatorname{tr}(B) = \operatorname{tr}(A)$, using the cyclic property of trace.
• Rank: $\operatorname{rank}(B) = \operatorname{rank}(A)$, because multiplication by invertibles preserves rank.
• Eigenvalues and characteristic polynomial: $\det(B - \lambda I) = \det(A - \lambda I)$.
• Minimal polynomial and Jordan form (covered in advanced courses).

Diagonalisation

A matrix $A$ is diagonalisable if it is similar to a diagonal matrix, that is, $A = P D P^{-1}$ for some diagonal $D$ and invertible $P$. In this case the columns of $P$ are eigenvectors of $A$ and the diagonal entries of $D$ are the corresponding eigenvalues. Diagonalisation is the cleanest similarity class: it makes powers easy ($A^k = P D^k P^{-1}$) and reveals the geometry of the transformation directly. Not every matrix is diagonalisable; the failure cases are captured by Jordan Normal Form.

The matrix widget above can illustrate similarity: set up a matrix $A$ with non-trivial geometry, then apply a coordinate change. The new matrix $B = P^{-1} A P$ has different entries but performs the same overall transformation, you can verify by checking that the determinant and trace remain unchanged.

• Markov chain stationary distributions: A transition matrix $P$ is often diagonalised to compute $P^n$ for large $n$, revealing the long-run stationary distribution as the eigenvector with eigenvalue 1. Without diagonalisation, computing $P^{1000}$ by hand is hopeless.
• Differential equations: The system $\dot{\mathbf{x}} = A \mathbf{x}$ has solution $\mathbf{x}(t) = e^{A t} \mathbf{x}_0$. If $A = P D P^{-1}$, then $e^{A t} = P e^{D t} P^{-1}$ and $e^{D t}$ is just exponentials on the diagonal, a closed-form solution.
• Quantum mechanics: Diagonalising the Hamiltonian operator in the energy basis reduces time evolution to multiplication by phase factors $e^{-i E_k t / \hbar}$