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Unformatted text preview: 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 and a nonzero vector v are referred to, respectively, as an eigenvalue and corresponding eigenvector for L if and only if L ( v ) = v . Equivalently, we can refer to an eigenvector v and its corresponding eigenvalue , or to the eigenpair (, 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 ) = = j . So, for this operator, i is an eigenvector with corresponding eigenvalue 1 , and j is an eigenvector with corresponding eigenvalue . ! Example 7.2: Suppose V is a twodimensional 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 bracketrightBiggbracketleftBigg 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 ( ) = = for every scalar . 10/10/2011 Eigenvectors and Hermitian Operators Chapter & Page: 72 ! Example 7.3: Let V be the vector space of all infinitelydifferentiable functions, and let be the differential operator ( f ) = f . Observe that ( sin ( 2 x )) = d 2 dx 2 sin ( 2 x ) = 4 2 sin ( 2 x ) . Thus, for this operator, 4 2 is an eigenvalue with corresponding eigenvector sin ( 2 x ) . 2 ? Exercise 7.1: Find other eigenpairs for . In practice, eigenvalues and eigenvectors are often associated with square matrices, with a scalar and a column matrix V being called an eigenvalue and corresponding eigenvector for a square matrix L if and only if LV = V . For example, in example 7.2 we saw that bracketleftBigg 1 2 3 2 bracketrightBiggbracketleftBigg 2 3 bracketrightBigg = 4 bracketleftBigg 2 3 bracketrightBigg . Thus, we would say that matrix bracketleftBigg 1 2 3 2 bracketrightBigg has eigenvalue = 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....
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This note was uploaded on 11/07/2011 for the course MA 607 taught by Professor Staff during the Summer '11 term at University of Alabama in Huntsville.
 Summer '11
 Staff
 Math, Eigenvectors, Vectors, Scalar, Vector Space

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