Consider n small beads each having mass m spaced at

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Unformatted text preview: is normal with nonzero eigenvalues {λ1 , λ2 , . . . , λr } , then the nonzero singular values of A are {|λ1 |, |λ2 |, . . . , |λr |}. • Right-hand and left-hand singular vectors for A are special eigenvectors for A∗ A and AA∗ , respectively. • Any complete orthonormal set of eigenvectors vi for A∗ A can serve as a complete set of right-hand singular vectors for A, and a corresponding complete set of left-hand singular vectors is given by ui = Avi / Avi 2 , i = 1, 2, . . . , r, together with any orthonormal basis {ur+1 , ur+2 , . . . , um } for N (A∗ ). Similarly, any complete orthonormal set of eigenvectors for AA∗ can be used as left-hand singular vectors for A, and corresponding right-hand singular vectors can be built in an analogous way. • The hermitian matrix B = D E T H IG R 0m×m A∗ A 0n×n of order m + n has nonzero eigenvalues {±σ1 , ±σ2 , . . . , ±σr } in which {σ1 , σ2 , . . . , σr } are the nonzero singular values of A. Y P Proof. Only the last point requires proof, and this follows by observing that if λ is an eigenvalue of B, then O C 0 A∗ A 0 x1 x2 =λ x1 x2 =⇒ Ax2 = λx1 A∗ x1 = λx2 =⇒ A∗ Ax2 = λ2 x2 , so each eigenvalue of B is the square of a singular value of A. But B is hermitian with rank (B) = 2r, so there are exactly 2r nonzero eigenvalues of B. Therefore, each pair ±σi , i = 1, 2, . . . , r, must be an eigenvalue for B. Example 7.5.4 Min-Max Singular Values. Since the singular values of A are the positive square roots of the eigenvalues of A∗ A, and since Ax 2 = (x∗ A∗ Ax)1/2 , it’s a corollary of the Courant–Fischer theorem (p. 550) that if σ1 ≥ σ2 ≥ · · · ≥ σn are the singular values for Am×n (n ≤ m), then σi = max dim V =i Copyright c 2000 SIAM min x∈V x 2 =1 Ax 2 and σi = min dim V =n−i+1 max x∈V x 2 =1 Ax 2 . Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 556 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 These expressions provide intermediate values between the extremes...
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