Matrix Similarity

In linear algebra, matrix similarity explains how two different matrices can represent the same linear transformation in different bases. The entries change with the coordinates, but the underlying transformation does not.

Two matrices $A, B \in \mathbb{K}^{n \times n}$ are said to be similar if there exists an invertible matrix $S \in \mathbb{K}^{n \times n}$ such that:

The matrix $S$ is the change-of-basis matrix connecting the two representations.

Basis Transformation and Coordinate Representation

Similar matrices represent the same linear transformation in two different bases. Let $e_1, \ldots, e_n \in \mathbb{K}^n$ be the canonical basis and $v_1, \ldots, v_n \in \mathbb{K}^n$ be another basis of $\mathbb{K}^n$.

If $S$ is an invertible matrix with columns $v_k$:

Then we have $v_k = S \cdot e_k$ or $e_k = S^{-1} \cdot v_k$ for $k = 1, \ldots, n$. The matrix $S$ represents the basis transformation.

A vector $x \in \mathbb{K}^n$ can be expressed in the canonical basis through coordinates $x_k$ and in the basis $v_1, \ldots, v_n$ through coordinates $\xi_k$:

The matrix $S^{-1}$ represents the coordinate transformation.

Linear Transformation in Different Bases

For the linear transformation $y = A \cdot x$, the canonical basis expresses $y$ with coordinates $y_k$. The basis $v_1, \ldots, v_n$ expresses the same vector with coordinates $\eta_k$:

The calculation gives:

The same coordinate relation can be written as follows.

In the basis $v_1, \ldots, v_n$, the linear transformation $y = A \cdot x$ is represented by $\eta = B \cdot \xi$ with the matrix:

Similar matrices represent one linear transformation with respect to different bases of $\mathbb{K}^n$.

Worked Basis Change Example

Let

The same transformation in the basis formed by the columns of $S$ has the matrix

Although the entries differ, both matrices have characteristic polynomial $(t-2)(t-3)$, trace $5$, and determinant $6$.

Invariant Properties of Similar Matrices

Because similar matrices describe the same transformation in different coordinates, they preserve its basis-independent properties. They have the same characteristic polynomial, determinant, trace, rank, eigenvalues with algebraic multiplicity, minimal polynomial, and Jordan block sizes.

They do not necessarily have the same entries, eigenvector coordinates, Euclidean norm, symmetry, or orthogonality. Those properties can depend on the chosen basis. A unitary or orthogonal change of basis preserves more geometric structure than a general similarity transformation.

Proof of Determinant Equality

For the determinant, we can show:

Since $\det(S^{-1}) = \frac{1}{\det(S)}$, then:

Eigenvalue Correspondence

If $v \in \mathbb{K}^n$ is an eigenvector of $A$ with eigenvalue $\lambda \in \mathbb{K}$, such that $A \cdot v = \lambda \cdot v$, then $w = S^{-1} \cdot v$ is an eigenvector of $B$ with the same eigenvalue:

Matrix similarity preserves the spectrum. When the basis changes, the eigenvector coordinates change from $v$ to $S^{-1}v$, but both coordinate vectors represent the same geometric direction.

Published: Jul 16, 2025. Updated: Sep 1, 2026.