However if each leading principal minor is positive

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Unformatted text preview: he first r left-hand singular vectors for A are uniquely determined by the first r right-hand singular vectors, while the last m − r left-hand singular vectors can be any orthonormal basis for N (A∗ ). If U is constructed from V as described above, then U is guaranteed to be unitary because for IG R U = u1 · · · ur |ur+1 · · · um = U1 |U2 and V = v1 · · · vr |vr+1 · · · vn = V1 |V2 , U1 and U2 each contain orthonormal columns, and, by using (7.5.9), Y P ∗ R (U1 ) = R AV1 D−1 = R (AV1 ) = R (AV1 D) = R [AV1 D][AV1 D] ⊥ ⊥ = R (AA∗ AA∗ ) = R (AA∗ ) = R (A) = N (A∗ ) = R (U2 ) . O C The matrix V is unitary to start with, but, in addition, R (V1 ) = R (V1 D) = R ([V1 D][V1 D]∗ ) = R (A∗ A) = R (A∗ ) and ⊥ R (V2 ) = R (A∗ ) = N (A). These observations are consistent with those established on p. 407 for any URV factorization. Otherwise something would be terribly wrong because an SVD is just a special kind of a URV factorization. Finally, notice that there is nothing special about starting with V to build a U —we can also take the columns of any unitary U that diagonalizes AA∗ as left-hand singular vectors for A and build corresponding right-hand singular vectors in a manner similar to that described above. Below is a summary of the preceding developments concerning singular values together with an additional observation connecting singular values with eigenvalues. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.5 Normal Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 555 Singular Values and Eigenvalues It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] For A ∈ C m×n with rank (A) = r, the following statements are valid. • The nonzero eigenvalues of A∗ A and AA∗ are equal and positive. • The nonzero singular values of A are the positive square roots of the nonzero eigenvalues of A∗ A (and AA∗ ). • If A...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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