On the nonuniqueness of the factorization factors in the product singular value decomposition

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On the nonuniqueness of the factorization factors in the product singular value decomposition

^聻

Delin Chu ^{a, ∗} , Bart De Moor ^b,1

a

Department of Mathematics, National University of Singapore, Lower Kent Road, Singapore 119260, Singapore

b

Department of Electrical Engineering (ESAT), Research Group SISTA, Katholieke Universiteit Leuven, Kardinaal Mercierlaan 94, B-3001 Leuven, Belgium

Received 18 December 1998; accepted 4 April 2000 Submitted by V. Mehrmann

Abstract

The product singular value decomposition (PSVD) of two matrices is revisited in this pa- per. The nonuniqueness of the factorization factors in the PSVD is characterized in a way different from that in existing work. © 2000 Elsevier Science Inc. All rights reserved.

AMS classification: 65F15; 65H15

Keywords: Product singular value decomposition; Nonuniqueness; Factorization factors

聻

This work is supported by several institutions:

1. The Flemish Government:

(a) Concerted Research Action GOA-MIPS (Model-based Information Processing Systems), (b) the FWO (Fund for Scientific Research—Flanders) project G.0292.95: Matrix algorithms and

differential geometry for adaptive signal processing, system identification and control,

(c) the FWO project G.0256.97: Numerical Algorithms for Subspace System Identification, Extension to Special Cases,

(d) the FWO Research Communities: ICCoS (Identification and Control of Complex Systems) and Advanced Numerical Methods for Mathematical Modelling.

2. The Belgian State, Prime Minister’s Office—Federal Office for Scientific, Technical and Cultural Affairs: Interuniversity Poles of Attraction Programme (IUAP P4-02 (1997–2001): Modeling, Iden- tification, Simulation and Control of Complex Systems; and IUAP P4-24 (1997–2001): Intelligent Mechatronic Systems (IMechS)).

∗

Corresponding author. Fax: +65-7795452.

E-mail addresses: matchudl@math.nus.edu.sg (D. Chu), Bart.Demoor@esat.kuleuven.ac.be (B. De Moor).

1

Tel.: +32-16-32-1970; fax: +32-16-32-1709.

0024-3795/00/$ - see front matter



2000 Elsevier Science Inc. All rights reserved.

PII: S 0 0 2 4 - 3 7 9 5 ( 0 0 ) 0 0 1 3 6 - 1

1. Introduction

The product singular value decomposition (PSVD) is a generalization for two matrices of the (ordinary) singular value decomposition (SVD) for one matrix. The explicit formulation of the PSVD was given for the first time by Fernando and Hammarling [7], who called it the
SVD. In this paper, unless noted, we always denote

r _a = rank(A), r _b = rank(B), r _ab = rank AB ^T

, X ^−T = (X ⁻¹ ) ^T for any given matrices A, B of appropriate dimensions and nonsingular matrix X.

Let us first state the following:

Theorem 1 (The PSVD Theorem). Given matrices A ∈ R ^m×n and B ∈ R ^p×n . Then there exist orthogonal matrices U ∈ R ^m×m , V ∈ R ^p×p and a nonsingular matrix X ∈ R ^n×n such that

U AX =





r _ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

r _ab S 0 0 0

r _a − r _ab 0 I 0 0

m − r a 0 0 0 0



, (1)

V BX ^−T =





r _ab r _a − r ab r _b − r ab n + r _ab − r a − r b

r _ab S 0 0 0

r _b − r _ab 0 0 I 0

p − r _b 0 0 0 0



,

where

S = diag{σ 1 I _i

1

, σ 2 I _i

2

, . . . , σ _k I _i

_k

}, σ 1 > σ 2 > · · · > σ _k > 0, X k j =1

i _j = r _ab .

Observe that

U AB ^T
V ^T =





r _ab r _b − r _ab p − r _b

r _ab SS ^T 0 0

r _a − r _ab 0 0 0

m − r _a 0 0 0



, hence, SS ^T contains the singular values of AB ^T .

Algorithmic ideas to implement the PSVD in a numerically reliable way can be

found in [2,7]. Applications include the orthogonal Procrustes problem [1], com-

puting balancing transformations for state space systems [7,9], and computing the

Kalman decomposition of a linear system [8]. The PSVD could also be applied in

the computation of approximate intersections between subspaces in the stochastic

realization problem [6], as an alternative to canonical correlation analysis.

The structure and geomety of the PSVD have been studied in [4]. In particular, the nonuniqueness of the factorization factors in the PSVD has been analyzed in detail.

In this paper, we revisit the PSVD. Our purpose is to characterize the nonuniqueness of the factors in the PSVD in a way different from that in [4].

2. Main result

Before we state our main result, we need a technical lemma.

Lemma 2. Given A ∈ R ^m×n , B ∈ R ^p×n . Then there exist orthogonal matrices U _a ∈ R ^m×m , V _b ∈ R ^p×p and Q _ab ∈ R ^n×n such that

U _a AQ _ab =





r ab r a − r ab r b − r ab n + r ab − r a − r b

r _ab A 11 A 12 0 0

r _a − r ab 0 A 22 0 0

m − r _a 0 0 0 0



, (2)

V _b BQ _ab =





r _ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

r ab B 11 0 B 13 B 14

r b − r ab 0 0 B 23 0

p − r b 0 0 0 0



,

where A 11 , A 22 , B 11 , B 23 are nonsingular.

Proof. See Appendix A. 

Based on Lemma 2, we can prove Theorem 1. In fact, we have:

Corollary 3. Given A ∈ R ^m×n and B ∈ R ^p×n . Let orthogonal matrices U a , V b and Q ab be defined as in Lemma 2. Assume that the SVD of A 11 B ₁₁ ^T is

U 11 A 11 B ₁₁ ^T V ₁₁ ^T = S ² , (3)

where U 11 , V 11 are orthogonal and S is defined as in Theorem 1. Define U =

 U 11

I _m−r

_ab



U _a , V =

 V 11

I _p−r

_ab

 V _b ,

X = Q _ab



 

 

 

A ⁻¹ ₁₁ U ₁₁ ^T S −A ⁻¹ ₁₁ A 12 A ⁻¹ ₂₂ 0 0

0 A ⁻¹ ₂₂ 0 0

B ₁₃ ^T V ₁₁ ^T S ⁻¹ 0 B ₂₃ ^T 0 B ₁₄ ^T V ₁₁ ^T S ⁻¹ 0 0 I



 

 

 

.

Then U AX and V BX ^−T are in the form (1).

Corollary 3 provides an alternative and very simple way to characterize the PSVD, which is different from the derivation in [4,5,7]. We are now in the position to present our main result.

Theorem 4 (Main Result). Given A ∈ R ^m×n and B ∈ R ^p×n . Let orthogonal matri- ces U _a , V _b , Q _ab , U 11 and V 11 be defined as in Lemma 2 and Corollary 3. Assume that U ∈ R ^m×m and V ∈ R ^p×p are orthogonal and X ∈ R ^n×n is nonsingular. Then U AX and V BX ^−T are in the form (1) if and only if

U =





r _ab r _a − r _ab m − r _a r ab P 11 U 11

r a − r ab P 22

m − r a P 33



U _a ,

V =





r ab r b − r ab p − r b

r ab P 11 V 11

r _b − r ab W 22

p − r _b W 33



V b , (4)

X =Q _ab



 

 

 



A ⁻¹ ₁₁ U ₁₁ ^T SP ₁₁ ^T −A ⁻¹ ₁₁ A 12 A ⁻¹ ₂₂ P ₂₂ ^T 0 0

0 A ⁻¹ ₂₂ P ₂₂ ^T 0 0

B ₁₃ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T X 32 B ₂₃ ^T W ₂₂ ^T X 34

B ₁₄ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T X 42 0 X 44



 

 

 

 ,

where P 11 , P 22 , P 33 , W 22 and W 33 are arbitrary orthogonal matrices, P 11 the block-diagonal:

P 11 =



 

 

i 1 i 2 · · · i k

i 1 P ₁₁ ⁽¹⁾ i 2 P ₁₁ ⁽²⁾

.. . . . .

i _k P ₁₁ ^(k)



 

  , (5)

X 32 , X 34 and X 42 the real and arbitrary, and X 44 is arbitrary real nonsingular matrix.

Obviously, Theorem 4 gives a complete characterization of the nonuniqueness

property of the factorization factors in the PSVD.

3. The proof of Theorem 4

In order to prove Theorem 4, we need two preliminary lemmas, in which the first one is well known and the second one is a direct consequence of QR and QL factorizations.

Lemma 5. The SVD of AB ^T is given by

 U 11

I _m−r

_ab

 U _a

 AB ^T

 V 11

I _p−r

_ab

 V _b

 T

=

 S ² 0

0 0

 . Furthermore, if orthogonal matrices U and V satisfy

U AB ^T V ^T =

 S ² 0

0 0

 , then

U =





r _ab r _a − r _ab m − r _a r ab P 11 U 11

r a − r ab P 22 P 23

m − r a P 32 P 33



U a ,

(6)

V =





r _ab r _b − r _ab p − r _b r _ab P 11 V 11

r _b − r _ab W 22 W 23

p − r _b W 32 W 33



V _b ,

where P 11 ,

 P 22 P 23

P 32 P 33

 and

 W 22 W 23

W 32 W 33



are orthogonal, and P 11 is of the block-diagonal form (5).

Proof. The proof is trivial. 

Lemma 6. Let X ∈ R ^n×n be nonsingular. Then there exists an orthogonal matrix Q such that

Q ^T X =



 



r ab r a − r ab r b − r ab n + r ab − r a − r b

r ab L 11 L 12 0 0

r a − r ab 0 L 22 0 0

r b − r ab L 31 L 32 L 33 L 34

n + r ab − r a − r b L 41 L 42 L 43 L 44



 

,

(7)

where

L 11 , L 22 and

 L 33 L 34

L 43 L 44



are nonsingular.

Proof. We partition X to be

X =  r ^ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

n X 1 X 2 X 3 X 4

 .

Let ˜ Q be such that

Q ˜ ^T 

X 3 X 4

 =



 



r _b − r _ab n + r _ab − r _a − r _b

r _ab 0 0

r _a − r _ab 0 0

r _b − r _ab L 33 L 34

n + r ab − r a − r b L 43 L 44



 

, and denote

Q ˜ ^T 

X 1 X 2

 =



 



r _ab r _a − r _ab

r ab ˜L 11 ˜L 12

r a − r ab ˜L ²¹ ˜L ²² r b − r ab L 31 L 32

n + r ab − r a − r b L 41 L 42



 

.

Now, there exists an orthogonal matrix ˆ Q satisfying

Q ˆ ^T  ˜ L 11 ˜L 12

˜L 21 ˜L 22



=

 r ^ab r a − r ab

r ab L 11 L 12

r a − r ab 0 L 22

 . Set

Q = ˜ Q

 Q ˆ I

 .

Then Q is such that (7) holds.  Now we prove Theorem 4.

Proof of Theorem 4. First we prove the sufficiency, then prove the necessity.

Sufficiency: Assume that U ∈ R ^m×m and V ∈ R ^p×p are orthogonal, X ∈ R ^n×n is nonsingular, and U AX and V BX ^−T are in the form (1). Then U AB ^T V ^T is the SVD of AB ^T , and hence, by Lemma 5, U and V are of the form (6). Note that

 0 P 32 P 33

 U a AX = 0, 

0 W 32 W 33

 V b BX ^−T = 0.

This implies that

 0 P 32 P 33

 U _a AQ _ab = 0, 

0 W 32 W 33

 V _b BQ _ab = 0.

So,

P 32 = 0, W 32 = 0. (8)

Consider that

 P 22 P 23

P 32 P 33

 and

 W 22 W 23

W 32 W 33



are orthogonal. Hence, we also have

P 23 = 0, W 23 = 0. (9)

By (8) and (9), we have that U and V are in the form (7).

According to Lemma 6, there exists an orthogonal matrix Q such that Q ^T X is of the form (7). Set

Q = 

Q 1 Q 2 Q 3 Q 4

 .

Then A 

Q 3 Q 4

 = 0. (10)

But, let us partition Q _ab in Lemma 2 into

Q _ab =  r ^ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b Q _ab1 Q _ab2 Q _ab3 Q _ab4 

. Then we know

A 

Q _ab3 Q _ab4 

= 0.

Hence,

 Q 3 Q 4

 = 

Q _ab3 Q _ab4  "

Q ˜ 33 Q ˜ 34

Q ˜ 43 Q ˜ 44

# ,

 Q 1 Q 2

 = 

Q _ab1 Q _ab2  "

Q ˜ 11 Q ˜ 12

Q ˜ 21 Q ˜ 22

#

with "

Q ˜ 11 Q ˜ 12

Q ˜ 21 Q ˜ 22

# and

"

Q ˜ 33 Q ˜ 34

Q ˜ 43 Q ˜ 44

#

orthogonal. Since we also have BQ 2 = 0, BQ _ab2 = 0, so,

Q ˜ 21 = 0, Q ˜ 12 = 0.

Hence, we have

Q = Q ab ×



 



r _ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

r _ab Q ˜ 11

r _a − r _ab Q ˜ 22

r _b − r _ab Q ˜ 33 Q ˜ 34

n + r _ab − r _a − r _b Q ˜ 43 Q ˜ 44



 



with

Q ˜ 11 , Q ˜ 22 and

"

Q ˜ 33 Q ˜ 34

Q ˜ 43 Q ˜ 44

#

orthogonal. We can write X to be

X =Q _ab



 

 Q ˜ 11

Q ˜ 22

Q ˜ 33 Q ˜ 34

Q ˜ 43 Q ˜ 44



 





 



L 11 L 12 0 0

0 L 22 0 0

L 31 L 32 L 33 L 34

L 41 L 42 L 43 L 44



 



=Q ab ×



 



r _ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

r ab X 11 X 12 0 0

r a − r ab 0 X 22 0 0

r b − r ab X 31 X 32 X 33 X 34

n + r ab − r a − r b X 41 X 42 X 43 X 44



 

.

Obviously,

X 11 , X 22 and

 X 33 X 34

X 43 X 44



are nonsingular. Now we have



 P 11 U 11

P 22

P 33







 A 11 A 12 0 0

0 A 22 0 0

0 0 0 0







 



X 11 X 12 0 0

0 X 22 0 0

X 31 X 32 X 33 X 34

X 41 X 42 X 43 X 44



 



=



 S 0 0 0

0 I 0 0

0 0 0 0



 (11)

and



 P 11 V 11

W 22

W 33







 B 11 0 B 13 B 14

0 0 B 23 0

0 0 0 0





=



 S 0 0 0

0 0 I 0

0 0 0 0







 



X 11 X 12 0 0

0 X 22 0 0

X 31 X 32 X 33 X 34

X 41 X 42 X 43 X 44



 



T

. (12)

A simple calculation yields that (11) and (12) hold if and only if P 11 U 11 A 11 X 11 = S,

P 22 A 22 X 22 = I, (13)

P 11 U 11 (A 11 X 12 + A 12 X 22 ) = 0 and

P 11 V 11 B 11 = SX ^T 11 , P 11 V 11 B 13 = SX ^T ₃₁ ,

W 22 B 23 = X ^T ₃₃ , (14)

P 11 V 11 B 14 = SX ^T ₄₁ , X ^T ₄₃ = 0.

Equivalently, (11) and (12) hold if and only if X 11 = A ⁻¹ ₁₁ U ₁₁ ^T SP ₁₁ ^T ,

X 22 = A ⁻¹ ₂₂ P ₂₂ ^T ,

X 12 = −A ⁻¹ ₁₁ A 12 A ⁻¹ ₂₂ P ₂₂ ^T ,

X 33 = B ₂₃ ^T W ₂₂ ^T , (15)

X 31 = B ₁₃ ^T V ₁₁ ^T P ₁₁ ^T S ⁻¹ = B ₁₃ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T , X 41 = B ₁₄ ^T V ₁₁ ^T P ₁₁ ^T S ⁻¹ = B ₁₄ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T , X 43 = 0.

Moreover, (15) also implies that X 32 , X 42 , X 34 and X 44 are arbitrarily, and X 44 is nonsingular because

 X 33 X 34

X 43 X 44



is nonsingular and X 43 = 0. Therefore, X is also in the form (4). Up to now, we have completed the proof of sufficiency in Theorem 4.

Necessity: Let orthogonal matrices U ∈ R ^m×m , V ∈ R ^p×p and nonsingular ma- trix X ∈ R ^n×n are in the form (4). Then a simple calculation gives that

U AX =



 P 11 SP ₁₁ ^T 0 0 0 0 P 22 P ₂₂ ^T 0 0

0 0 0 0



 =



 S 0 0 0

0 I 0 0

0 0 0 0



 (16)

and



 S 0 0 0

0 0 I 0

0 0 0 0







 

 



A ⁻¹ ₁₁ U ₁₁ ^T SP ₁₁ ^T −A ⁻¹ ₁₁ A 12 A ⁻¹ ₂₂ P ₂₂ ^T 0 0

0 A ⁻¹ ₂₂ P ₂₂ ^T 0 0

B ₁₃ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T X 32 B ₂₃ ^T W ₂₂ ^T X 34

B ₁₄ ^T V ₁₁ ^T S ⁻¹ P ₁₁ ^T X 42 0 X 44



 

 



T

=



 SP 11 SU 11 A ^−T ₁₁ 0 SP 11 S ⁻¹ V 11 B 13 SP 11 S ⁻¹ V 11 B 14

0 0 W 22 B 23 0

0 0 0 0





=



 P 11 V 11 B 11 0 P 11 V 11 B 13 P 11 V 11 B 14

0 0 W 22 B 23 0

0 0 0 0





=



 P 11 V 11

W 22

W 33







 B 11 0 B 13 B 14

0 0 B 23 0

0 0 0 0



 , (17)

equivalently, we have

V BX ^−T =



 S 0 0 0

0 0 I 0

0 0 0 0



 . (18)

In (17), we have used the following equalities:

SP 11 = P 11 S, S ^T = S, V 11 B 11 = S ² U 11 A ^−T ₁₁ . Hence, the “necessity” follows directly from (16) and (18). 

4. Conclusion

In this paper, the nonuniqueness of the factorization factors in the PSVD has been characterized in a way different from that in [4].

Appendix A

Before we prove Lemma 2, we need to recall the QR factorization with column pivoting and URV decomposition [1], which will be the building blocks of our con- structive proof of Lemma 2.

It is well known that any matrix A ∈ R ^m×n can be factorized as

U A =

 R 1 R 2

0 0



P, (A.1)

where U and P are orthogonal matrix and permutation matrix, respectively, R 1 is nonsingular and upper triangular. The factorization (A.1) is called the QR factoriza- tion of A with column pivoting.

If we continue to squeeze [ R

1

R

2

] into upper triangular form by applying a se- quence of Householder transformations, then we have the following URV decompo- sition of A, i.e., we get an orthogonal matrix V such that

U AV =

 R 0

0 0



(A.2) with R nonsingular and upper triangular.

Now we are ready to present a constructive proof for Lemma 2.

Proof of Lemma 2. We prove Lemma 2 constructively by following six steps:

Step 1: Compute the U RV decomposition of A:

ˆU 1 A ˆ Q 1 =

" r ^a n − r _a r _a A ⁽¹⁾ ₁ 0 m − r _a 0 0

# ,

where A ⁽¹⁾ ₁ is nonsingular.

Step 2: Compute the QR factorization of B ˆ Q 1 with column pivoting:

ˆV 1 (B ˆ Q 1 ) =

" r ^a n − r a

r b B ₁ ⁽¹⁾ B ₃ ⁽¹⁾ p − r _b 0 0

# , where 

B

1⁽¹⁾

B

3⁽¹⁾

 is of full row rank. Note that r ab = rank AB ^T

= rank 

A ⁽¹⁾ ₁ B ₁ ⁽¹⁾

 = rank  B ₁ ⁽¹⁾

 . Step 3: Compute the QR factorization of (B ₁ ⁽¹⁾ ) ^T :

B ₁ ⁽¹⁾ Q ˆ 2 = h r ^ab r _a − r ab

r _b B ₁ ⁽²⁾ 0 i with B ₁ ⁽²⁾ of full column rank. Set

A ⁽¹⁾ ₁ Q ˆ 2 = h r ^ab r a − r ab

r a A ⁽²⁾ ₁ A ⁽²⁾ ₂ i

.

Step 4: Compute the QR factorizations of A ⁽²⁾ ₁ and B ₁ ⁽²⁾ with column pivoting:

ˆU 3

h

A ⁽²⁾ ₁ A ⁽²⁾ ₂ i =

 r ^ab r _a − r _ab r _ab A 11 A 12

r _a − r _ab 0 A 22



,

ˆV 3

h

B ₁ ⁽²⁾ B ₃ ⁽¹⁾ i =

" r _ab n − r _a r _ab B 11 B ₁₃ ⁽³⁾ r _b − r _ab 0 B ₂₃ ⁽³⁾

# ,

where A 11 , A 22 and B 11 are nonsingular. We also have that B ₂₃ ⁽³⁾ is of full row rank.

Step 5: Compute the QR factorization of (B ₂₃ ⁽³⁾ ) ^T with pivoting:

"

B ₁₃ ⁽³⁾ B ₂₃ ⁽³⁾

# Q ˆ 4 =

 r ^b − r _ab n + r _ab − r _a − r _b

r _ab B 13 B 14

r _b − r _ab B 23 0

 ,

where B 23 is nonsingular.

Step 6: Set U _a =  ˆ U 3

I m−r

a

 ˆU 1 ,

V _b =  ˆ V 3

I _p−r

_b

 ˆV 1 ,

Q _ab = ˆ Q 1

 Q ˆ 2

I _n−r

_a

  I _r

_a

Q ˆ 4

 . Now we have

U _a AQ _ab =





r _ab r _a − r ab r _b − r ab n + r _ab − r a − r b

r _ab A 11 A 12 0 0

r _a − r _ab 0 A 22 0 0

m − r _a 0 0 0 0



,

V _b BQ _ab =





r _ab r _a − r _ab r _b − r _ab n + r _ab − r _a − r _b

r _ab B 11 0 B 13 B 14

r _b − r _ab 0 0 B 23 0

p − r _b 0 0 0 0





with A 11 , A 22 , B 11 and B 23 nonsingular. Therefore, Lemma 2 follows.  In general, the size of A 11 (i.e., the size of B 11 ) is much smaller than those of A and B. Moreover, the condensed form (2) can be computed via numerically sta- ble ways. Hence, similar to [3], the condensed form (2) can be considered to be an efficient preprocessing algorithm for computing the PSVD of matrix pair (A, B).

This preprocessing algorithm will reduce the complexity of the Kogbetliantz-type

algorithm in [2]. Therefore, the PSVD of (A, B) can be computed in the following

two phases:

• Reduce matrix pair (A, B) to the condensed form (2).

• Compute the PSVD of matrix pair (A 11 , B 11 ) using the Kogbetliantz-type algo- rithm in [2].

References

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