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7330hw1

# 7330hw1 - Math 7330 Functional Analysis Homework 1 Fall...

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Math 7330: Functional Analysis Fall 2002 Homework 1 A. Sengupta In the following, V is a finite-dimensional complex vector space with a Hermitian inner-product ( · , · ), and A : V V a linear map. 1. Let e 1 , ..., e n be an orthonormal basis of V . (i) Show that the matrix for A relative to the basis e 1 , ..., e n has A ij = ( Ae j , e i ) as the entry at the i –th row and j –th column. (ii) Show that for the matrix of A * , ( A * ) ij = A ji 1

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2. Suppose that A is a normal operator, i.e. it commutes with its adjoint: AA * = A * A Show that | Ax | = | A * x | for all x V . 3. Show that for a complex number λ C the following are equivalent: A - λI is not invertible there is a non-zero vector x V for which Ax = λx det( A - λI ) = 0 2
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If k C and non-zero y V satisfy Ay = ky then k is an eigenvalue of A and y is an eigenvector corresponding to the eigenvalue k . In general, we shall use the notation M k = { v V : Av = kv } = ker( A - kI ) The set of all λ C for which A - λI is not invertible is called the spectrum of A .
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7330hw1 - Math 7330 Functional Analysis Homework 1 Fall...

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