Similarity &
Change of Basis

Understand how similar matrices represent the same linear transformation in different bases, and master change-of-basis computations using $ P^{-1}AP $.

Linear Algebra

What Is a Basis?

A basis for a vector space $ V $ is a set of vectors $ \{ \mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n \} $ that is both linearly independent and spans $ V $. Every vector in $ V $ can be written as a unique linear combination of the basis vectors.

Linearly Independent

No vector in the set can be written as a combination of the others.

Spanning

Every vector in $ V $ is reachable via combinations of the basis vectors.

Coordinate Vectors

Given a basis $ \mathcal{B} = \{ \mathbf{b}_1, \mathbf{b}_2 \} $, the coordinate vector of $ \mathbf{x} $ relative to $ \mathcal{B} $ is the column $ [\mathbf{x}]_\mathcal{B} $ of scalars such that:

$$ \mathbf{x} = c_1 \mathbf{b}_1 + c_2 \mathbf{b}_2 \;\Longrightarrow\; [\mathbf{x}]_\mathcal{B} = \begin{pmatrix} c_1 \\ c_2 \end{pmatrix} $$

The Change-of-Basis Matrix

Let $ \mathcal{B} = \{ \mathbf{b}_1, \dots, \mathbf{b}_n \} $ be a basis for $ \mathbb{R}^n $. The change-of-basis matrix from $ \mathcal{B} $ to the standard basis $ \mathcal{E} $ is:

$$ P = \begin{pmatrix} \mathbf{b}_1 & \mathbf{b}_2 & \cdots & \mathbf{b}_n \end{pmatrix} $$

It converts $ \mathcal{B} $-coordinates to standard coordinates:

$$ \mathbf{x} = P \, [\mathbf{x}]_\mathcal{B} $$

Similar Matrices: Definition

Two $ n \times n $ matrices $ A $ and $ B $ are similar if there exists an invertible matrix $ P $ such that:

$$ B = P^{-1} A P $$

Geometrically, $ A $ and $ B $ represent the same linear transformation $ T $, but expressed in different bases. $ A $ uses the standard basis; $ B $ uses the basis whose columns form $ P $.

Properties Preserved by Similarity

Similar matrices share many fundamental properties because they represent the same transformation. If $ B = P^{-1}AP $, then $ A $ and $ B $ have the same:

Eigenvalues

Same characteristic polynomial $ \det(A - \lambda I) = \det(B - \lambda I) $.

Determinant

$ \det(B) = \det(P^{-1})\det(A)\det(P) = \det(A) $.

Trace

$ \operatorname{tr}(B) = \operatorname{tr}(P^{-1}AP) = \operatorname{tr}(A) $.

Similar vs. Other Equivalences

Relation	Form	Preserves
Similar	$ P^{-1}AP $	Eigenvalues, det, trace, rank
Row Equivalent	$ EA $ (elem. row ops)	Row space, rank (not eigenvalues)
Congruent	$ P^{T}AP $	Signature (for symmetric $ A $)
Unitarily Similar	$ U^{\ast}AU $, $ U $ unitary	All of similarity + singular values

Unpacking $ B = P^{-1}AP $

Read the formula $ B = P^{-1}AP $ right-to-left as a three-step pipeline:

1

Convert coordinates: Multiply by $ P $ to translate from $ \mathcal{B} $-coordinates to standard coordinates.

2

Apply the transformation: Multiply by $ A $ in standard coordinates.

3

Convert back: Multiply by $ P^{-1} $ to return the result to $ \mathcal{B} $-coordinates.

How to Compute a Change of Basis

1

Identify the new basis $ \mathcal{B} $: Write down the basis vectors $ \mathbf{b}_1, \mathbf{b}_2, \dots $

2

Form $ P $: Place the basis vectors as columns: $ P = [\mathbf{b}_1 \mid \mathbf{b}_2 \mid \cdots] $

3

Compute $ P^{-1} $: Use row reduction on $ [P \mid I] $ or the formula for $ 2\times 2 $ inverses.

4

Compute $ B = P^{-1}AP $: Multiply in order: first $ AP $, then $ P^{-1}(AP) $.

5

Verify: Check that $ A $ and $ B $ have the same eigenvalues, determinant, and trace.

Example 1: Basic Similarity

Let $ A = \begin{pmatrix} 4 & -2 \\ 1 & 1 \end{pmatrix} $ and $ \mathcal{B} = \left\{ \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 2 \\ 1 \end{pmatrix} \right\} $. Find $ B = P^{-1}AP $.

Example 2: Verifying Similarity

Are $ A = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix} $ and $ B = \begin{pmatrix} 3 & 0 \\ -2 & 1 \end{pmatrix} $ similar?

1

Check eigenvalues: $ A $ is upper triangular, so $ \lambda_A = 1, 3 $. Similarly, $ B $ is lower triangular, so $ \lambda_B = 3, 1 $. ✓ Same set.

2

Check trace: $ \operatorname{tr}(A) = 1 + 3 = 4 $, $ \operatorname{tr}(B) = 3 + 1 = 4 $. ✓ Match.

3

Check determinant: $ \det(A) = 3 $, $ \det(B) = 3 $. ✓ Match.

4

Conclusion: Necessary conditions are met. Since both are $ 2 \times 2 $ with the same distinct eigenvalues, they are diagonalizable to the same diagonal matrix, hence similar.

Example 3: Finding $ P $

Find an invertible $ P $ such that $ P^{-1}AP = D $ where $ A = \begin{pmatrix} 5 & 4 \\ 1 & 2 \end{pmatrix} $.

Why Diagonalize?

The most powerful application of similarity is diagonalization. If $ A = PDP^{-1} $ where $ D $ is diagonal, then computing powers becomes trivial:

$$ A^k = P D^k P^{-1} = P \begin{pmatrix} \lambda_1^k & 0 \\ 0 & \lambda_2^k \end{pmatrix} P^{-1} $$

Matrix Powers

$ A^{100} $ reduces to computing $ \lambda_i^{100} $ individually.

Differential Equations

$ e^{At} = P \operatorname{diag}(e^{\lambda_i t}) P^{-1} $ for solving systems.

Recurrences

Fibonacci and similar sequences: closed-form via diagonalization.

Between Two Non-Standard Bases

To convert from basis $ \mathcal{B} $ to basis $ \mathcal{C} $ (neither being the standard basis), compose the change-of-basis matrices:

$$ P_{\mathcal{C} \leftarrow \mathcal{B}} = P_{\mathcal{C}}^{-1} \, P_{\mathcal{B}} $$

Common Mistakes

✕ Wrong Order

Writing $ PAP^{-1} $ instead of $ P^{-1}AP $. The inverse goes on the left.

✕ Rows vs. Columns

Placing basis vectors as rows of $ P $. They must be columns.

✕ Same Invariants = Similar

Having the same eigenvalues, trace, and det is necessary but not sufficient in general (e.g., non-diagonalizable matrices).

Summary of Key Ideas

Concept	Key Formula	Interpretation
Change-of-basis matrix	$ P = [\mathbf{b}_1 \mid \cdots \mid \mathbf{b}_n] $	Columns are new basis vectors
Coordinate conversion	$ \mathbf{x} = P[\mathbf{x}]_\mathcal{B} $	$ P $ maps $ \mathcal{B} $-coords to standard
Similarity	$ B = P^{-1}AP $	Same transformation, different basis
Diagonalization	$ A = PDP^{-1} $	$ P $ = eigenvectors, $ D $ = eigenvalues
Matrix power	$ A^k = PD^kP^{-1} $	Raise diagonal entries to power $ k $

Question 1 of 5

If $ B = P^{-1}AP $, which of the following is always true?

Similar matrices share the same characteristic polynomial, hence the same eigenvalues. However, their eigenvectors generally differ (they are related by $ P $), and $ P $ need not be orthogonal.

Question 2 of 5

Given $ P = \begin{pmatrix} 1 & 3 \\ 2 & 5 \end{pmatrix} $, what is $ P^{-1} $?

For a $ 2\times 2 $ matrix, $ P^{-1} = \frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix} $. Here $ \det(P) = 5 - 6 = -1 $, so $ P^{-1} = \frac{1}{-1}\begin{pmatrix} 5 & -3 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} -5 & 3 \\ 2 & -1 \end{pmatrix} $.

Question 3 of 5

If $ A $ is diagonalizable with $ A = PDP^{-1} $, what is $ A^3 $?

$ A^3 = (PDP^{-1})^3 = PDP^{-1} \cdot PDP^{-1} \cdot PDP^{-1} = PD^3P^{-1} $ because the inner $ P^{-1}P = I $ cancels. Choice (c) is technically equal to (a) but claims it "cannot simplify," which is false. Choice (d) forgets the $ P $ factors.

Question 4 of 5

Which pair of matrices is not similar?

$ \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix} $ is not diagonalizable (it's a Jordan block with eigenvalue 1), while $ I $ is already diagonal. If they were similar, $ P^{-1}\begin{pmatrix}1&1\\0&1\end{pmatrix}P = I $ would imply $ \begin{pmatrix}1&1\\0&1\end{pmatrix} = I $, a contradiction.

Question 5 of 5

In the formula $ B = P^{-1}AP $, the columns of $ P $ are:

The change-of-basis matrix $ P $ is formed by placing the new basis vectors as columns. Each column $ \mathbf{b}_i $ is written in standard coordinates. This is the fundamental definition of the change-of-basis matrix.

Similarity &
Change of Basis

What Is a Basis?

Coordinate Vectors

The Change-of-Basis Matrix

Similar Matrices: Definition

Properties Preserved by Similarity

Similar vs. Other Equivalences

Unpacking \( B = P^{-1}AP \)

How to Compute a Change of Basis

Example 1: Basic Similarity

Example 2: Verifying Similarity

Example 3: Finding \( P \)

Why Diagonalize?

Between Two Non-Standard Bases

Common Mistakes

Summary of Key Ideas

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Relation	Form	Preserves
Similar	\( P^{-1}AP \)	Eigenvalues, det, trace, rank
Row Equivalent	\( EA \) (elem. row ops)	Row space, rank (not eigenvalues)
Congruent	\( P^{T}AP \)	Signature (for symmetric \( A \))
Unitarily Similar	\( U^{\ast}AU \), \( U \) unitary	All of similarity + singular values

Concept	Key Formula	Interpretation
Change-of-basis matrix	\( P = [\mathbf{b}_1 \mid \cdots \mid \mathbf{b}_n] \)	Columns are new basis vectors
Coordinate conversion	\( \mathbf{x} = P[\mathbf{x}]_\mathcal{B} \)	\( P \) maps \( \mathcal{B} \)-coords to standard
Similarity	\( B = P^{-1}AP \)	Same transformation, different basis
Diagonalization	\( A = PDP^{-1} \)	\( P \) = eigenvectors, \( D \) = eigenvalues
Matrix power	\( A^k = PD^kP^{-1} \)	Raise diagonal entries to power \( k \)

Similarity &Change of Basis

What Is a Basis?

Coordinate Vectors

The Change-of-Basis Matrix

Similar Matrices: Definition

Properties Preserved by Similarity

Similar vs. Other Equivalences

Unpacking \( B = P^{-1}AP \)

How to Compute a Change of Basis

Example 1: Basic Similarity

Example 2: Verifying Similarity

Example 3: Finding \( P \)

Why Diagonalize?

Between Two Non-Standard Bases

Common Mistakes

Summary of Key Ideas

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Similarity &
Change of Basis