of the

i

.

as e a

L.

OTATION AND BREVIATIONS.

atrices are denote

are nonnegative intege

98 BART L. R. DE MOOR

the OSVD of the matrix AB’BA’ to the eigenvdue decomposition of the matrices B’BA’A and A’AB’B. We shall ako prove a lemma that permits us to express the PSVD of the matrix pair A, B in terms of their OSVDs when

AB’ = 0. In Section 2.2, we shall provide a variational characterization of the PSVD.

2.1. A Conslructi~ Proof

of the

THEOREM 1 (The PSVD). Every pair

of ^red

A = Q&J’, B = u&J-‘.

?A = r*-::

ln - r@#

%=

rb-::

P - 'b

where S, is spume diagonal with positive e&men& and tl = dAB’).

While some related eigenvalue problems were discussed in [13] and [16], the explicit fknulation of the PSVD is in Theomm 1 was given for the first time by Fernando and Hammarling in [8], who called it the IISVD.”

0 81 also a constructive proof was provided. It is however based on a Senma &mma 1 in [8D of which the proof is not correct. To give a counterexample to the proof, consider the pair A- ’

Q; gs&,.ps

SINGULAR VALUE DECOMPOSITION 99

Throughout the paper, we shall se the matrix Y defined as y = XBt [8], the factorization is present rent form, where a factorization of X is used. this may be preferable in analysing nu&uZ issues related to the , such an additional factorization is not relevant for our present purpose, which is the detailed exploration of

structural and geometrical pairs of nonzero

elements of S,,, and S, in their product the ~o&c~ si

structural information th There are four

possibilities: There are rl pairs of the form (6, &I with corresponding product singular value oi, i = 1, . . . . _ t”r. By convention, they are ordered so that niaq+r. There are r= - rr pairs (1,O) with corresponding product singular value 0. There are rb - f, pairs (0,l) with corresponding product singular value 0. There are n - t-a - rl, + t, pairs (O,O), which we shall call the trivial product singular value pairs, in analogy with the trivial quotient singular value pairs [4].

In the constructive proof of Theorem 1, we shall need the following four lemmas:

LEMMA 1. kt the OSW

of

matrix

given

Then the set of sol&ms

of

matrix

is

The first term is nothing else than AtB, where At is the Moore-Penrose pseudoinverse of A. It is also the unique minimum Frobenius norm solution.

RecaU that At is the Moore-Penrose inverse of A if it is the unique sohrtion T = A+ of

ATA= A, (0

TAT = T, (9

(AT)‘= AT, (3)

This pair of matrices satisfies the condition required by the lemma in [8] that AB” is diagonal.

With the notation of [8], we have that i = 1, j = = 2, r = 5. ile the proof of the lemma tates that f - i-j , because for our examp

ience, the proof of

= 1 t ?

“a2 ⁹

=

P ⁹

e theorem

s,’

eorem is to

.

I = t

I.

co

= 2’

. S

the expression

for

ence, the column vectors of Y2 belong to the null space of B’B,A!A, an rankY2=rankU,=r,-r,.

PfOOf fir Y3: Assume that Y3 = AtU4 + Vaz =VazT4. It follows that BtBA’AY3 = BtBAtAV,,T, = 0. This implies that the column vectors of Y3 belong to the null space of B ‘BAtA, and obviously rank Y3 = ran

if T4 is nonsingular.

Finally, we have to verify that with fixed U,, ^{U,, U4,} an always chose T3 and T4 to make the matrix

Y=(Y, Y2

y,)=(&

of fdl rank. Rewrite (111, using the OS

e conse ce 0 is:

.

( ;

C

eigenvectors of

consists 0 ree steps:

st we’

e

plies

is no

1

, 1

ecause

-1 .

11% 2

--I an iI is e upper r,Xr,

ran z2

ica s

ra a= - r, = r0 - q, ran

and Corollary 2, it follows that at can be related to the

= x(

sgq@,

t us now derive a variational interpreta- owing optimizath pr43bh:

ver vectors x a

7

s x* a _1’

-“(s;s,)x-‘,

’ diagonalizes the matrix for a pair of matrices is

tion for a pair

ces are equal, we

ient transformation). contragredi-

= T’GT is real diago-

. -

is

of.

c

en

ut the t sfomation

ave more 620

0 b ⁼

0

= c

ow assu

0

rest of Sections 3.2

e a commoil

again er t

s of A ha X rob) and

J ₁ S ha

zzz .

t = _.

ien ation

= r’

i

“I f-OF, - “‘I, “I, - f-1

-b-#,+-to "b-+I

es

of

- 2 =

f-0 -rl’

9

= 9

.

me

I 1 ^.

2 = .

the s ces

= -l/2

31 3 ’

-1- -

32 = + -I-

= -1 + + ;

-1 -1

+ =

2’

r1 ra-r1 ‘ab - ra - ‘1, + ‘1 ‘b - ‘I n - rah

‘1 ra-r1

=

rah - ra --I,+'1

ra-rl

=

“ah - ro - ^{3-1, C ri}

ra - r1

fl

= . f - *I n-r

r1

=

rl,-“n P - ‘h

emn must

t?SS . S

. .

=i t ^{1 ?2}

0

five submatrices for X aud

structure as in (71)

s XT and YR, where T and eorem 3, solve the set of

. TATION OF THE STRUCTURE

all relate the structure of the contragredient tion, to the geometry of of and be as in (26)

es

canonical angles

r _c2 ^{= r,}

fferent from etween the ranges of Va2,

Now consider e partitioning of and repeated here for convenience:

*a- fl ‘ah - ra - ‘/I + ‘1

rl ra-rl '018 - *a --pB+*1

Y as derived in Section 3, w

‘0 - t1 n - ‘aI,

X )

>

32 4

‘II - r1 n - ‘ah

1 2 31 32

1/e 1

w”nich is not orthogonal to the

so m (77):

and

ace 0

= .

oreover,

1 form an underdetermined l-(36) one can apply etermined equations as

= -1

1 al

s-a/z

1 1

S e co S

is is a se Si IiCeS

1 = ,

7)

1

= .

7) to er wi

exist matrices Z,Y, Zy, of appropriate

is

1 = _{1 al} s-l 2+

(

= (1

f these expressions are premu ^Wi near

equations for Z,Y:

e first factor 0

90 . .

tten more

ions are croci

r 3y is given by

ressions for Val Z;

wever, it will now requirement,

ces 8 constrai

ations (1 m5) xmit us to simplifj: uation (133) as

ow use Squations (126) an 7) to get

uation (128) that

( 137)

that ensures the o

is the product of the least squares

2

3 4

5 6

7 8

9 10 11 12 13

14

15 16

7

Golub and C. F. Van Loan, trk Computations, North Oxford Academic,

value decomposition of a product of two matrices, S 8, No. 4 (1986).

eere and L. Silverman,

18

c. e.

e decomposition, S t.

ah-ices,