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EigenHerm - 7 Eigenvectors and Hermitian Operators 7.1...

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7 Eigenvectors and Hermitian Operators 7.1 Eigenvalues and Eigenvectors Basic Definitions Let L be a linear operator on some given vector space V . A scalar λ and a nonzero vector v are referred to, respectively, as an eigenvalue and corresponding eigenvector for L if and only if L ( v ) = λ v . Equivalently, we can refer to an eigenvector v and its corresponding eigenvalue λ , or to the eigen-pair (λ, v ) . Do note that an eigenvector is required to be nonzero. 1 An eigenvalue, however, can be zero. ! Example 7.1: If L is the projection onto the vector i , L ( v ) = −→ pr i ( v ) , then L ( i ) = −→ pr i ( i ) = i = 1 · i and L ( j ) = −→ pr i ( j ) = 0 = 0 · j . So, for this operator, i is an eigenvector with corresponding eigenvalue 1 , and j is an eigenvector with corresponding eigenvalue 0 . ! Example 7.2: Suppose V is a two-dimensional vector space with basis A = { a 1 , a 2 } , and let L be the linear operator whose matrix with respect to A is L = bracketleftBigg 1 2 3 2 bracketrightBigg . Letting v = 2 a 1 + 3 a 2 , we see that | L ( v ) ) = L | v ) = bracketleftBigg 1 2 3 2 bracketrightBigg bracketleftBigg 2 3 bracketrightBigg = bracketleftBigg 2 + 6 6 + 6 bracketrightBigg = bracketleftBigg 8 12 bracketrightBigg = 4 bracketleftBigg 2 3 bracketrightBigg = 4 | v ) = | 4 v ) . This shows that L ( v ) = 4 v , so v is an eigenvector for L with corresponding eigenvalue 4 . 1 This simply is to avoid silliness. After all, L ( 0 ) = 0 = λ 0 for every scalar λ . 10/10/2011
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Eigenvectors and Hermitian Operators Chapter & Page: 7–2 ! Example 7.3: Let V be the vector space of all infinitely-differentiable functions, and let be the differential operator ( f ) = f ′′ . Observe that ( sin ( 2 π x )) = d 2 dx 2 sin ( 2 π x ) = − 4 π 2 sin ( 2 π x ) . Thus, for this operator, 4 π 2 is an eigenvalue with corresponding eigenvector sin ( 2 π x ) . 2 ? Exercise 7.1: Find other eigenpairs for . In practice, eigenvalues and eigenvectors are often associated with square matrices, with a scalar λ and a column matrix V being called an eigenvalue and corresponding eigenvector for a square matrix L if and only if LV = λ V . For example, in example 7.2 we saw that bracketleftBigg 1 2 3 2 bracketrightBigg bracketleftBigg 2 3 bracketrightBigg = 4 bracketleftBigg 2 3 bracketrightBigg . Thus, we would say that matrix bracketleftBigg 1 2 3 2 bracketrightBigg has eigenvalue λ = 4 and corresponding eigenvector V = bracketleftBigg 2 3 bracketrightBigg . This approach to eigenvalues and eigenvectors is favored by instructors and texts whose main interest is in getting their students to compute a bunch of eigenvalues and eigenvectors. Unfortunately, it also obscures the basis-independent nature of the theory, as well as the basic reasons we may be interested in eigen-thingies. It also pretty well limits us to the cases where the operators are only defined on finite-dimensional vector spaces. Admittedly, every linear operator on a finite-dimensional space can be described in terms of a square matrix with respect to some basis, and every square matrix can be viewed as a matrix with respect to some basis for some linear operator on some finite-dimensional vector space. So associating eigenpairs with matrices could
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