The same linear transformation has different matrix representations in different bases. Changing the basis changes the numbers but not the map itself. The relationship between two matrix representations of the same transformation is similarity — and choosing the right basis is how difficult matrices become simple ones.
The Problem
A linear transformation T:V→V is a fixed geometric object — it sends each vector to a definite image regardless of how coordinates are assigned. But the matrix that represents T depends on the choice of basis. Different bases assign different coordinates to the same vectors, and the matrix that converts input coordinates to output coordinates changes accordingly.
This raises a natural question: if T has matrix A in one basis and matrix A′ in another, how are A and A′ related? The answer is the similarity relation A′=P−1AP, where P is the change-of-basis matrix. Understanding this relation is the key to choosing bases strategically — picking the basis that makes the matrix as simple as possible.
The Change-of-Basis Matrix
If B and C are two bases for V, the change-of-basis matrix PC←B converts B-coordinates to C-coordinates:
[v]C=PC←B[v]B
Column j of P is the C-coordinate vector of the j-th basis vector of B. The reverse conversion uses the inverse: PB←C=P−1.
Worked Example
In R2, let B={(1,1),(1,−1)} and let C be the standard basis. The C-coordinates of the B-basis vectors are just their components: (1,1) and (1,−1). So
P=(111−1)
To find the B-coordinates of v=(5,1): solve Pc=(5,1). Using P−1=−21(−1−1−11), we get c=(3,2). So v=3(1,1)+2(1,−1).
The Similarity Relation
If T:V→V has matrix A in basis B and matrix A′ in basis C, then
A′=P−1AP
where P=PC←B is the change-of-basis matrix from B to C.
The derivation is direct. For any vector v, the transformation in B-coordinates reads [T(v)]B=A[v]B. Converting to C-coordinates: [T(v)]C=P−1[T(v)]B=P−1A[v]B=P−1AP[v]C. Since this holds for every v, the matrix of T in basis C is P−1AP.
Two matrices related by A′=P−1AP for some invertible P are called similar. Similarity is an equivalence relation: every matrix is similar to itself (P=I), similarity is symmetric (A′=P−1AP implies A=PA′P−1), and it is transitive.
Properties Preserved by Similarity
Similar matrices represent the same transformation, so they share every property that is intrinsic to the transformation rather than to a particular coordinate system.
The determinant is preserved: det(P−1AP)=det(P−1)det(A)det(P)=det(A).
The trace is preserved: tr(P−1AP)=tr(APP−1)=tr(A) by the cyclic property.
The eigenvalues are preserved: det(P−1AP−λI)=det(P−1(A−λI)P)=det(A−λI), so the characteristic polynomial — and therefore all eigenvalues with their multiplicities — is the same.
The rank is preserved: multiplying by invertible matrices cannot change the rank.
Individual matrix entries, symmetry, and sparsity are generally not preserved. A symmetric matrix A can become non-symmetric under P−1AP if P is not orthogonal.
Diagonalization as a Change of Basis
If T has nlinearly independent eigenvectors v1,…,vn with eigenvalues λ1,…,λn, use them as the basis B. In this eigenvector basis, T acts by scaling each basis vector:
T(vi)=λivi
The matrix of T in this basis is diagonal: D=diag(λ1,…,λn).
The change-of-basis matrix P has the eigenvectors as columns: P=[v1⋯vn]. The similarity relation gives A=PDP−1, or equivalently D=P−1AP.
Diagonalization is the most powerful application of basis change. It reduces matrix powers to diagonal powers: Ak=PDkP−1=Pdiag(λ1k,…,λnk)P−1. It simplifies differential equations, recurrence relations, and any computation involving repeated application of the same transformation.
When Diagonalization Fails
Not every matrix is diagonalizable. A transformation may not have n linearly independent eigenvectors — this happens when the geometric multiplicity of some eigenvalue is strictly less than its algebraic multiplicity.
For example, A=(2012) has eigenvalue λ=2 with algebraic multiplicity 2, but the eigenspace is one-dimensional (spanned by (1,0)). There is no basis of eigenvectors, so A cannot be diagonalized.
In such cases, the best achievable form under similarity is the Jordan normal form: a block-diagonal matrix where each block is an upper triangular matrix with a single eigenvalue on the diagonal and ones on the superdiagonal. The Jordan form is unique up to ordering of blocks and is the canonical representative of the similarity class. Its full development belongs to advanced linear algebra.
Orthogonal Similarity
When the change-of-basis matrix P is orthogonal (P−1=PT), the similarity relation becomes A′=PTAP. This is called orthogonal similarity.
Orthogonal similarity preserves more than ordinary similarity. If A is symmetric, then PTAP is also symmetric — a property that ordinary similarity does not guarantee.
The Spectral Theorem states that every real symmetric matrix is orthogonally similar to a diagonal matrix. The eigenvectors of a symmetric matrix can be chosen orthonormal, and the columns of P form an orthonormal basis. This is a stronger conclusion than ordinary diagonalizability — the diagonalizing basis is not just independent but orthonormal, which simplifies projections, least squares, and numerical computation.
Worked Example: Full Basis Change
Let A=(4213). Find a diagonalization A=PDP−1.
The characteristic polynomial is det(A−λI)=(4−λ)(3−λ)−2=λ2−7λ+10=(λ−2)(λ−5). Eigenvalues: λ1=2, λ2=5.
For λ1=2: (A−2I)v=0 gives (2211)v=0, so v1=(1,−2).
For λ2=5: (A−5I)v=0 gives (−121−2)v=0, so v2=(1,1).
P=(1−211),D=(2005)
Verification: P−1=31(12−11), and PDP−1=(1−211)(2005)31(12−11)=(4213)=A.
The standard basis is the default, but it is rarely the best choice for a given problem.
An eigenvector basis diagonalizes the matrix, reducing powers and exponentials to operations on diagonal entries. A system of differential equations x′=Ax decouples into independent scalar equations when A is diagonal.
An orthonormal basis simplifies projections and least-squares computations. Coordinates relative to an orthonormal basis are computed by dot products rather than by solving systems, and numerical errors are minimized because the change-of-basis matrix has condition number 1.
A Jordan basis achieves the simplest possible form for non-diagonalizable matrices, isolating the defective eigenvalues into small blocks.
Choosing the right basis is often the key insight that converts a hard problem into an easy one. The transformation does not change — only its numerical description does — but the right description can make all the difference between a tractable computation and an intractable one.