Similar Matrices — Definition, Formula & Examples

Similar matrices are square matrices

A

and

B

where

B = P^{-1}AP

for some invertible matrix

P

. They represent the same linear transformation but expressed in different bases.

Two $n \times n$ matrices $A$ and $B$ are similar if there exists an invertible $n \times n$ matrix $P$ such that $B = P^{-1}AP$ . Similarity is an equivalence relation on the set of $n \times n$ matrices: it is reflexive, symmetric, and transitive. Similar matrices share the same eigenvalues, determinant, trace, rank, and characteristic polynomial.

Key Formula

B = P^{-1}AP

Where:

$A$ = An $n \times n$ matrix
$B$ = An $n \times n$ matrix similar to $A$
$P$ = An invertible $n \times n$ matrix (the change-of-basis matrix)

How It Works

To show two matrices are similar, you need to find an invertible matrix

P

that conjugates one into the other. In practice, you often check necessary conditions first: if

A

and

B

have different eigenvalues, traces, or determinants, they cannot be similar. If those match, you attempt to find

P

by solving

AP = PB

for the columns of

P

. Diagonalization is a special case — when you diagonalize

A

into

D = P^{-1}AP

, you are showing

A

is similar to a diagonal matrix

D

Worked Example

Problem: Verify that A and B are similar, where

A = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}

B = \begin{pmatrix} 3 & 0 \\ 1 & 2 \end{pmatrix}

, and

P = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Step 1: Compute

P^{-1}

. Since

P

swaps rows, it is its own inverse.

P^{-1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

Step 2: Compute

AP

AP = \begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix}

Step 3: Compute

P^{-1}AP

P^{-1}AP = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 1 & 2 \end{pmatrix} = B

Answer: Since

P^{-1}AP = B

, the matrices

A

and

B

are similar. Notice they share the same trace (

5

), determinant (

6

), and eigenvalues (

2

and

3

Why It Matters

Similar matrices appear throughout linear algebra whenever you change basis — for instance, diagonalizing a matrix to compute its powers efficiently. In differential equations and control theory, recognizing similarity lets you simplify a system into a canonical form without changing its fundamental behavior.

Common Mistakes

Mistake: Assuming that equal eigenvalues, trace, and determinant guarantee similarity.

Correction: These are necessary but not sufficient conditions. For example, the

2 \times 2

identity matrix and

\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

share eigenvalues, trace, and determinant, but they are not similar.