III.86 The Spectrum

G. R. Allan

In the theory of LINEAR MAPS [I.3 §4.2], or operators, on a VECTOR SPACE [I.3 §2.3], the notions of EIGENVALUE AND EIGENVECROR [I.3 §4.3] play an important role. Recall that if V is a vector space (over Image or Image) and if T : VV is a linear mapping, then an eigenvector of T is a nonzero vector e in V such that T(e) = λe for some scalar λ then λ is the eigenvalue corresponding to the eigenvector e. If V is finite dimensional, then the eigenvalues are also the roots of the characteristic polynomial χ(t) = det(tIT

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