Transcription of Chapter 7 Canonical Forms - Duke University
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Chapter 7 Canonical Forms - Duke University
Chapter 7 Canonical Eigenvalues and EigenvectorsDefinition a vector space over the fieldFand letTbe a linearoperator onV. AneigenvalueofTis a scalar Fsuch that there exists a non-zero vectorv VwithTv= v. Any vectorvsuch thatTv= vis called aneigenvectorofTassociated with the eigenvalue value .Definition (T)of a linear operatorT:V Vis the set ofall scalars such that the operator(T I)is not `2be the Hilbert space of infinite square-summable se-quences andT:V Vbe the right-shift operator defined byT(v1,v2,..) = (0,v1,v2,..).SinceTis not invertible, it follows that the scalar0is in the spectrum ofT. But, itis not an eigenvalue becauseTv= 0impliesv= 0and an eigenvector must be anon-zero vector. In fact, this operator does not have any finite-dimensional spaces, things are quite a bit the matrix representation of a linear operator on a finite-dimensional vector spaceV, and let be a scalar.
Definition 7.1.5. Let be an eigenvalue of the matrix A. The eigenspace associated with is the set E = fv 2VjAv = vg. The algebraic multiplicity of is the multiplicity of the zero at t= in the characteristic polynomial ˜ A(t). The geometric multiplicity of an eigenvalue is equal to dimension of the eigenspace E or nullity(A tI). Theorem 7.1.6.
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