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sol65 - Partial Solution Set Leon 6.5 6.5.1 We are to show...

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Partial Solution Set, Leon § 6.5 6.5.1 We are to show first of all that A and A T have the same nonzero singular values, then to describe the relationship between the singular value decompositions of A and A T . So begin by assuming that σ is a nonzero singular value for A . By definition of singular value, we know that λ = σ 2 is a positive eigenvalue of A T A . Let x be an eigenvector for A T A belonging to λ . Then A T A x = λ x . So λ ( A x ) = A ( λ x ) = A ( A T A x ) = AA T ( A x ) , so A x is an eigenvector for AA T , also belonging to λ . Conversely, suppose that σ is a nonzero singular value for A T . Then λ = σ 2 is a positive eigenvalue for AA T , with eigenvector x , i.e., AA T x = λ x . Then λ ( A T x ) = A T ( λ x ) = A T ( AA T x ) = A T A ( A T x ) , so A T x is an eigenvector for A T A , also belonging to λ . Thus AA T and A T A have the same positive eigenvalues, hence the same nonzero singular values. How, then, are the singular value decompositions for A and A T related? This is more easily answered: if A = U Σ V T , then A T = ( U Σ V T ) T = V Σ T U T . 6.5.2 We are to find the singular value decompositions of several matrices, using the method outlined in the text. Here are two of the solutions. (a) A = 1 1 2 2 . We begin by finding A T A = 5 5 5 5 . The eigenvalues are λ 1 = 10 and λ 2 = 0. So σ 1 = 10, and σ 2 = 0, and we have Σ = 10 0 0 0 . We know that V is a diagonalizing matrix for A T A
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