Linear Algebra › Section E · Eigenvalues and eigenvectors › Topic E.4
Eigenvectors and linear transformations
A linear transformation does not come with one fixed matrix: its matrix depends on the coordinates you measure it in. Choosing a basis of eigenvectors gives the simplest matrix of all, and explains what the factorisation A = PDP⁻¹ really says.
In this topic
- One transformation, many matrices
- The matrix relative to two bases
- Worked example: evaluating a polynomial
- Transformations from a space to itself
- Eigenvectors of a transformation
- Changing the basis of Rⁿ
- The diagonal matrix representation
- Worked example: two bases for one map
- Similar matrices, same transformation
- Common mistakes to avoid
- Key takeaways
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