Similar matrices have the same eigenvalues is a fundamental result in linear algebra that connects the concepts of similarity transformations and spectral properties of matrices. Understanding why this equality holds provides insight into how changing bases affects (or does not affect) the intrinsic characteristics of a linear transformation, and it underpins many practical techniques such as diagonalization, stability analysis, and numerical computations.
What Are Similar Matrices?
Two square matrices (A) and (B) of the same size are said to be similar if there exists an invertible matrix (P) such that
[ B = P^{-1}AP . ]
The matrix (P) represents a change of basis, and the similarity transformation (P^{-1}AP) re‑expresses the linear transformation represented by (A) in a new coordinate system. Because similarity only reshapes the representation without altering the underlying transformation, many intrinsic properties—such as determinant, trace, rank, and, importantly, eigenvalues—remain unchanged.
Eigenvalues and the Characteristic Polynomial
An eigenvalue (\lambda) of a matrix (A) is a scalar for which there exists a non‑zero vector (v) (the eigenvector) satisfying
[ Av = \lambda v . ]
Equivalently, (\lambda) is a root of the characteristic polynomial of (A), defined as
[ p_A(\lambda) = \det(\lambda I - A) . ]
The eigenvalues are precisely the values that make this polynomial zero. Because of this, if two matrices share the same characteristic polynomial, they must have identical eigenvalues (including algebraic multiplicities) Turns out it matters..
Why Similar Matrices Share Eigenvalues
Proof via Determinant Properties
Assume (B = P^{-1}AP) with (P) invertible. The characteristic polynomial of (B) is
[ \begin{aligned} p_B(\lambda) &= \det(\lambda I - B) \ &= \det\bigl(\lambda I - P^{-1}AP\bigr) . \end{aligned} ]
Insert the identity (I = P^{-1}P) inside the determinant:
[ \lambda I - P^{-1}AP = P^{-1}(\lambda I)P - P^{-1}AP = P^{-1}(\lambda I - A)P . ]
Using the multiplicative property of determinants and the fact that (\det(P^{-1}) = 1/\det(P)),
[ \begin{aligned} p_B(\lambda) &= \det\bigl(P^{-1}(\lambda I - A)P\bigr) \ &= \det(P^{-1}),\det(\lambda I - A),\det(P) \ &= \frac{1}{\det(P)},\det(\lambda I - A),\det(P) \ &= \det(\lambda I - A) \ &= p_A(\lambda) . \end{aligned} ]
Thus (p_B(\lambda) = p_A(\lambda)) for every (\lambda), which means (A) and (B) have exactly the same eigenvalues, each with the same algebraic multiplicity.
Alternative View: Change of Basis
If (v) is an eigenvector of (A) with eigenvalue (\lambda), then
[ Av = \lambda v . ]
Multiplying on the left by (P^{-1}) gives
[ P^{-1}Av = \lambda P^{-1}v . ]
Since (B = P^{-1}AP), we can rewrite the left‑hand side as
[ B(P^{-1}v) = \lambda (P^{-1}v) . ]
Hence (P^{-1}v) is an eigenvector of (B) associated with the same eigenvalue (\lambda). This shows that similarity merely re‑labels eigenvectors while preserving eigenvalues.
Relationship Between Eigenvectors
While eigenvalues are invariant, eigenvectors transform according to the change‑of‑basis matrix. In real terms, specifically, if (v) is an eigenvector of (A), then (w = P^{-1}v) is an eigenvector of (B) for the same eigenvalue. Conversely, any eigenvector (w) of (B) yields an eigenvector (v = Pw) of (A). This one‑to‑one correspondence preserves the geometric multiplicity (the dimension of the eigenspace) as well, because the map (v \mapsto P^{-1}v) is a linear isomorphism.
Illustrative Examples
Example 1: A Simple 2×2 Similarity
Let
[ A = \begin{pmatrix} 4 & 1 \ 0 & 2 \end{pmatrix}, \qquad P = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}, \qquad P^{-1} = \begin{pmatrix} 1 & -1 \ 0 & 1 \end{pmatrix}. ]
Compute
[ B = P^{-1}AP = \begin{pmatrix} 1 & -1 \ 0 & 1 \end{pmatrix} \begin{pmatrix} 4 & 1 \ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}
\begin{pmatrix} 4 & 0 \ 0 & 2 \end{pmatrix}. ]
Both (A) and (B) are upper‑triangular, so their eigenvalues are the diagonal entries: (4) and (2). Indeed, despite the different forms, the eigenvalues coincide.
Example 2: Diagonalization
If a matrix (A) is diagonalizable, there exists an invertible (P) such that
[ D = P^{-1}AP ]
is a diagonal matrix whose diagonal entries are the eigenvalues of (A). The similarity transformation explicitly exhibits the eigenvalues, confirming that any matrix similar to a diagonal matrix shares those same diagonal values as its eigenvalues The details matter here..
Example 3: Non‑Diagonalizable but Similar
Consider
[ A = \begin{pmatrix} 1 & 1 \ 0 & 1 \end{pmatrix}, \qquad P = \begin{pmatrix} 2 & 0 \ 0 & 1 \end{pmatrix}, \qquad P^{-1} = \begin{pmatrix}
Since
[ P^{-1}= \begin{pmatrix} \tfrac12 & 0 \[2pt] 0 & 1 \end{pmatrix}, ]
we obtain
[ B = P^{-1}AP = \begin{pmatrix} \tfrac12 & 0 \[2pt] 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1 \[2pt] 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \[2pt] 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & \tfrac12 \[2pt] 0 & 1 \end{pmatrix}. ]
Both (A) and (B) are Jordan blocks with the sole eigenvalue (1); the only distinction is the magnitude of the super‑diagonal entry. Consequently they possess the same eigenvalue (1) with algebraic multiplicity 2 and geometric multiplicity 1, confirming that similarity leaves the size of Jordan blocks — and hence the eigen‑structure — unchanged And that's really what it comes down to..
Because a Jordan block of size (k) has minimal polynomial ((x-\lambda)^{k}), the minimal polynomial of (A) and of (B) is identical. In fact, every polynomial that annihilates (A) also annihilates (B) and vice‑versa, since similarity preserves the underlying linear map up to a change of basis. And other quantities that are invariant under similarity include the trace, determinant, rank, and the dimensions of eigenspaces (the geometric multiplicities). These invariants are encoded in the Jordan canonical form, which is unique up to the ordering of the Jordan blocks.
Boiling it down, while the explicit eigenvectors are re‑labelled by the change‑of‑basis matrix (P), the spectrum — the set of eigenvalues together with their algebraic and geometric multiplicities — remains exactly the same for any two similar matrices. This invariance underlies the classification of linear operators up to similarity and explains why similarity transformations are the natural notion of “sameness’’ in linear algebra.
