MAT223H1 Lecture 14: Lecture 14 3.2(3)
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Recall t is nonzero and eigenvectorfora with eigenvalue . Remark if a matrix has no real eigenvalues then a is invertible. Example the rotation matrix at has no real eigenvalues and is therefore invertible. Why is this thm true let"s argue that a is invertible if 0 is an eigenvalue. Suppose o is an eigenvalue thenthere is a nonzero. Thus is nonzero andin nulla1 so nullah it by invertible matrixthru. Now suppose a is not invertible thenbytheinvertible matrixthru nullah . Thereforehereis a nonzero ie at so i ist ai 0 is an eigenvector ofa with eigenvalue 0 t n is theset ofvectors i. Defthereignspace ofa corresponding thefixed value ofm denote theespace byen at afn for. If a is nxn then each eigenspace of a is a subspace of rn and i are in ei then aput at . Problem findthe eigenspau. tn corresponding to is inevery eigenspace. ir.
