Linear Algebra · Advanced · 7 min
Diagonalization
An eigenvector basis turns a linear transformation into separate coordinate scalings.
WHAT HAPPENS IF…Do repeated eigenvalues always prevent diagonalization?
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For the symmetric family, normalized eigenvectors (1,1)/√2 and (1,−1)/√2 give Q with Q⁻¹=Qᵀ. In that basis A scales by m+k and m−k. Repeated eigenvalues at k=0 are harmless. A Jordan block lacks enough eigenvectors; a quarter-turn lacks a real eigenbasis.
A detail that matters
The criterion is a full eigenvector basis, not simply having real eigenvalues.
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