Page 466 - Advanced Linear Algebra

450 Advanced Linear Algebra

(~ > ?

Find ( b . Hint : Is it better to work with ( i ( or ( ( i ?
d
4. Let ? ~ ²%%Ä % ³ ! be a column matrix over . Find a singular-value

decomposition of .
?
5. Let (  4 Á ²-³ and let )  4 b Á b ²-³ be the square matrix
(
)~ > i ?
(
block
Show that, counting multiplicity, the nonzero eigenvalues of ) are
precisely the singular values of together with their negatives. Hint : Let
(
i
(~ < ' < be a singular-value decomposition of ( and try factoring )

<
into a product <:< i where is unitary. Do not read the following second
hint unless you get stuck. Second Hint : Verify the block factorization
< ' i < i
)~ > ? > ? > i ?
< ' <
( b
What are the eigenvalues of the middle factor on the right? Try b
and b . c )
6. Use the results of the previous exercise to show that a matrix
i
²-³, its adjoint ( , its transpose ( and its conjugate ( all have
!
( 4 Á
<
the same singular values. Show also that if and < Z are unitary, then (
and <(< Z have the same singular values.
7. Let ( 4 ²-³ be nonsingular. Show that the following procedure

produces a singular-value decomposition (~ < ' < i of .
(
)
a Write (~ <+< i where + ~ diag² Á Ã Á ³ and the 's are
positive and the columns of < form an orthonormal basis of
eigenvectors for . We never said that this was a practical procedure.)
(
(
b Let ' ) diag ~ ° Á Ã Á ² ° ³ where the square roots are nonnegative.

i
Also let <~ < and U ~ ( <' c .

(
8. If (~ ² ³ is an d matrix, then the Frobenius norm of is
Á
°
)) ~ - 8 Á 9
(
Á
Show that )) ~ is the sum of the squares of the singular values of
(
-
(.