Linear Algebra 61 | Similar Matrices

TL;DR

Two matrices A and B are similar when an invertible S exists with A = S^{-1}BS, representing the same linear map in different coordinates.

Briefing Cornell Notes

Briefing

Similar matrices are the algebraic way to say two matrices represent the same linear map in different coordinate systems: if there exists an invertible matrix S such that A = S^{-1}BS (equivalently S^{-1}BS = A), then A and B are similar. This invertible change of basis matters because it preserves core structural features of the associated linear maps. In particular, injectivity is identical for the two matrices: A is injective if and only if B is injective. The same kind of invariance also holds for surjectivity, reflecting that the kernel and image dimensions behave consistently under similarity.

The most important spectral consequence is that similar matrices share the same characteristic polynomial. For a matrix A, the characteristic polynomial is P_A(λ) = det(A − λI), a degree-n polynomial. Using the similarity relation A = S^{-1}BS, the determinant expression can be rewritten by expressing the identity I as S^{-1}S, then factoring out S^{-1} on the left and S on the right. Because determinants are multiplicative, the determinant contributions from S^{-1} and S cancel out (det(S^{-1})det(S) = det(I) = 1), leaving det(B − λI). As a result, P_A(λ) = P_B(λ), so A and B have exactly the same eigenvalues, with the same algebraic multiplicities.

The discussion then connects these invariance results to eigenvector structure. While the eigenvalues and their algebraic multiplicities stay fixed under similarity, the eigen spaces themselves can change after the coordinate transformation. The dimensions of the eigenspaces (geometric multiplicities) are also preserved, but the actual subspaces in the ambient vector space may be different because the change of basis reshapes how vectors are represented.

From there, the transcript moves to a major payoff: for certain classes of matrices, similarity can be used to simplify computation dramatically. Every normal matrix is similar to a diagonal matrix. That includes self-adjoint (Hermitian) matrices, which are normal. In this diagonal form, the diagonal entries are precisely the eigenvalues of the original matrix, repeated according to their algebraic multiplicities—so a normal n×n matrix becomes diagonal with n eigenvalue entries counting multiplicity. Diagonal matrices are far easier to work with than general matrices because many operations (powers, functions of the matrix, solving linear systems) reduce to elementwise operations on the diagonal.

For general square matrices, the strong diagonalization guarantee fails. Instead, one can only ensure a weaker canonical form: similarity to an upper triangular matrix. The eigenvalues still appear on the diagonal with the correct algebraic multiplicities, but there may be nonzero entries above the diagonal. The transcript points to Jordan normal form as the more precise framework for understanding this remaining structure, while noting that the next step before fully using these canonical forms is learning how to compute eigenvectors—since the transformation matrix S and its inverse are tied directly to eigenvectors.

Cornell Notes

Two square matrices A and B are similar if an invertible matrix S exists with A = S^{-1}BS. Similarity corresponds to the same linear map written in different coordinates, so key properties like injectivity (and surjectivity) are preserved. Most importantly, similarity leaves the characteristic polynomial unchanged: det(A − λI) equals det(B − λI), so A and B have the same eigenvalues with the same algebraic multiplicities. Geometric multiplicities (dimensions of eigenspaces) also remain the same, though the eigenspaces themselves can move under the change of basis. This preservation of spectral data motivates diagonalization: every normal matrix is similar to a diagonal matrix, while a general matrix is only guaranteed to be similar to an upper triangular one (with Jordan form describing the finer structure).

What does it mean for two matrices to be similar, and why must S be invertible?

Matrices A and B are similar if there exists an invertible matrix S such that A = S^{-1}BS. Invertibility is essential because S represents a legitimate change of basis: it must map vectors bijectively so that the coordinate transformation can be reversed. Without invertibility, the relationship would not preserve the underlying linear-map structure in a reversible way.

How does similarity guarantee that A and B have the same characteristic polynomial?

Similarity gives A = S^{-1}BS. In P_A(λ) = det(A − λI), substitute A to get det(S^{-1}BS − λI). Rewrite λI as S^{-1}(λI)S (equivalently insert I = S^{-1}S) so the determinant becomes det(S^{-1}(B − λI)S). Using multiplicativity of determinants, det(S^{-1})det(B − λI)det(S) = det(B − λI) because det(S^{-1})det(S) = det(I) = 1. Therefore P_A(λ) = P_B(λ).

What spectral information is preserved under similarity, and what can still change?

Similarity preserves the eigenvalues and their algebraic multiplicities because the characteristic polynomial is unchanged. It also preserves the dimensions of eigenspaces (geometric multiplicities). However, the eigenspaces themselves can change as subspaces because the change of basis S^{-1} reshapes how eigenvectors are represented in the ambient space.

Why are normal matrices singled out for diagonalization?

Normal matrices (including self-adjoint/Hermitian matrices) are guaranteed to be similar to a diagonal matrix. In that diagonal form, the diagonal entries are the eigenvalues of the original matrix, repeated according to algebraic multiplicities. This turns many computations into simple diagonal operations.

What is the weaker guarantee for an arbitrary square matrix?

For a general square matrix, diagonalization is not always possible. The guaranteed canonical form is similarity to an upper triangular matrix. The eigenvalues still appear on the diagonal with the correct algebraic multiplicities, but nonzero entries may remain above the diagonal. Jordan normal form refines this by describing the block structure behind those off-diagonal entries.

Review Questions

If A = S^{-1}BS, show directly (using determinant properties) why det(A − λI) = det(B − λI).
Which quantities are preserved under similarity: eigenvalues, algebraic multiplicities, geometric multiplicities, and eigenspaces themselves?
Why does diagonalization hold for normal matrices but not for arbitrary matrices? What form replaces it?

Key Points

1
Two matrices A and B are similar when an invertible S exists with A = S^{-1}BS, representing the same linear map in different coordinates.
2
Similarity preserves injectivity and surjectivity of the associated linear maps.
3
The characteristic polynomial is invariant under similarity: det(A − λI) = det(B − λI).
4
As a consequence, similar matrices have identical eigenvalues with the same algebraic multiplicities.
5
Geometric multiplicities (dimensions of eigenspaces) are preserved under similarity, even though the eigenspaces themselves may change.
6
Every normal matrix is similar to a diagonal matrix, with eigenvalues on the diagonal repeated by algebraic multiplicity.
7
For general matrices, similarity guarantees only an upper triangular form, motivating Jordan normal form for the remaining structure.

Highlights

Similarity is defined via an invertible change of basis: A = S^{-1}BS.

A determinant manipulation shows the characteristic polynomial stays the same under similarity, so eigenvalues and algebraic multiplicities match exactly.

Normal matrices are always diagonalizable up to similarity; self-adjoint matrices fall into this case.

General matrices may fail to diagonalize, but they can always be put into an upper triangular form where eigenvalues sit on the diagonal.