MATH 01210 · Linear maps & eigenstuff

Eigenvalues / eigenvectors

Definition

Av=λv with v≠0: v is an eigenvector, λ an eigenvalue. Solve det(A−λI)=0 for the characteristic polynomial; then (A−λI)v=0 for eigenvectors.

Eigenspace for λ is N(A−λI) (plus {0} convention as a subspace).

Eigenvalues can be complex even for real matrices.

Trace = sum of eigenvalues; det = product (over algebraically closed fields, counting multiplicity).

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