What is it
Undergraduate eigenvector work extends into eigenspaces (the full set of eigenvectors for a given eigenvalue, plus the zero vector) and their role in determining whether a matrix can be diagonalised.
Why it matters
Eigenvectors and their spaces are central to principal component analysis in data science, vibration modes in engineering, and quantum states in physics — a concept that recurs across nearly every quantitative discipline.
Exam tip
When an eigenvalue is repeated, always check the dimension of its eigenspace against its multiplicity in the characteristic equation — this comparison is exactly what determines whether the matrix is diagonalisable.
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