← MATH 01210 · Linear Algebra
MATH 01210 · Linear maps & eigenstuff
Diagonalization idea
Definition
A is diagonalizable if A=PDP⁻¹ for diagonal D (eigenvalues) and P (eigenvectors as columns). Equivalent: enough independent eigenvectors to form a basis.
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Powers: Aᵏ = P Dᵏ P⁻¹ — cheap once diagonalized.
n distinct eigenvalues ⇒ diagonalizable; repeated eigenvalues may fail if geometric multiplicity is too small.
Not every matrix is diagonalizable (Jordan form generalizes).
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1.Decide if [[1,1],[0,1]] is diagonalizable.