Change of basis & similar matrices
A vector’s coordinates depend on the basis, and so does the matrix of a linear map. If P has the new basis vectors as columns, then P converts new coordinates to old, and P⁻¹ goes the other way. Two matrices representing the same map in different bases are similar: B = P⁻¹AP — which is why similar matrices share eigenvalues, trace, determinant and rank.
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Method: how to approach it
The order below is what examiners expect to see, and each step carries its own marks.
- Build the change of basis matrixColumns of P are the new basis vectors written in the old coordinates. Then [v]_old = P[v]_new.
- Convert coordinatesTo go from old to new, multiply by P⁻¹. Getting the direction backwards is the standard error.
- Convert the matrix of a mapIf A is the matrix in the old basis, the new one is P⁻¹AP.
- Use the invariants as a checkSimilar matrices have the same trace, determinant, rank and characteristic polynomial. A mismatch means an error.
Worked example
The vector v has coordinates (3, 1) in the basis {(1,1), (1,−1)}. Find its standard coordinates.
- P has the new basis vectors as columns: P = [[1, 1], [1, −1]].
- [v]_standard = P·[v]_new.
- = [[1,1],[1,−1]]·(3,1).
- First component 3 + 1 = 4; second 3 − 1 = 2.
Answer. v = (4, 2) in standard coordinates — that is, 3(1,1) + 1(1,−1) = (4,2) ✓.
Where marks get dropped
These are the specific errors that cost credit on change of basis & similar matrices questions — QED's rubric penalises each of them separately.
- Using P where P⁻¹ is needed. P takes NEW coordinates to OLD; the reverse direction needs the inverse.
- Writing the new basis vectors as rows of P instead of columns.
- Confusing similarity with equality. Similar matrices represent the same map but are generally different matrices.
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Change of basis & similar matrices — frequently asked questions
Why do similar matrices share eigenvalues?
Because det(P⁻¹AP − λI) = det(P⁻¹(A − λI)P) = det(A − λI). The characteristic polynomial is unchanged, so the eigenvalues are too.
What is diagonalisation in this language?
Choosing a basis of eigenvectors. In that basis the map is represented by a diagonal matrix, so A = PDP⁻¹ is a change of basis.
Are all matrices with equal eigenvalues similar?
No. [[1,1],[0,1]] and the identity both have eigenvalue 1 twice but are not similar — the Jordan form distinguishes them.
The rest of Linear Algebra
Matrices, systems, determinants, eigenvalues. Each subtopic below has its own method, worked example and mark-losing traps.
- 1Matrix arithmetic & inverses
- 2Gaussian elimination
- 3Determinants
- 4Vector spaces, span & linear independence
- 5Basis & dimension
- 6Eigenvalues & eigenvectors
- 7Linear maps & their matrices
- 8Rank, nullity & the rank–nullity theorem
- 9Column space, null space & solution sets
- 10Orthogonality, projections & Gram–Schmidt
- 11Diagonalisation & powers of a matrix
- 12Dot product, norms & angles
- 13Change of basis & similar matrices
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