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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
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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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