vol3_pp103-118. 161KB Jun 04 2011 12:06:14 AM

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The Electronic Journal of Linear Algebra.

A publication of the International Linear Algebra Society. Volume 3, pp. 103-118, June 1998.

ISSN 1081-3810. http://math.technion.ac.il/iic/ela

ELA

ON THE RODMAN-SHALOMCONJECTURE REGARDING THE

JORDANFORM OFCOMPLETIONS OF PARTIALUPPER

TRIANGULARMATRICES

CRISTINA JORDANy, JUAN R. TORREGROSAy,

AND ANA M. URBANO y Dedicated to Hans Schneider on the occasion of his seventieth birthday. Abstract. Rodman and Shalom [LinearAlgebra anditsApplications, 168:221{249, 1992] con-jecture the following statement: LetAbe a lower irreducible partial upper triangularnnmatrix overF such thattr ace(A) = 0. Letn

1 n

2

np1 be a set ofppositive integers such that p

X i=1 n

i=

n. There exists a nilpotent completionA c of

Awhose Jordan form consists ofpblocks of sizesn

i n

i,

i= 1;2;:::;pif and only ifr(A k)

X i:n

i k

(n i

,k),k= 1;2;:::;n 1.

In this paper this conjecture is solved in two cases: when the minimal rank ofA is 2, and for matrices of size 55.

Keywords. Completion problems, Jordan structure, Minimal rank, Majorization. AMSsubject classications. 15A18, 15A21

1. Intro duction. We consider nn matrices A = [a

ij] over an innite eld F. A matrix is said to be a partial matrix if some of its entries are given elements

from the eldF, while the rest of them can be arbitrarily chosen and treated as free

independent variables. If those last elements are xed, the resultant matrix is called acompletionA

c of the matrix

A. The completion problems consist of nding all the

completions of a given partial matrix with prescribed properties. We are interested in

partial upper triangular matrices, namely, partial matricesA= [a

ij] where a

ij, ij,

are the given elements. For this kind of matrices some completion problems have been studied. For example, problems related to the rank of the matrix can be found in [9], to eigenvalues in [1, 9], to Jordan form in [2, 7, 8, 9] and to controllability of linear systems in [3, 4].

LetA be a partial matrix. We denote by r(A) the minimal rank of all possible

completions ofA, that is,r(A) = min A

c

r ank(A

c) where the minimum is taken over

the set of all the ranks of possible completions ofA. Moreover, for any positive integer k, we denote r(A

k) = min A

c

r ank((A c)

k), where the minimum is taken over the set

of all the ranks of the kth power of possible completions ofA.

We recall that a matrix A is lower similar to a matrixB (we denote A B)

if there exists a lower triangular matrix S such that A = SBS

,1. In addition, a

matrixAof sizennis said to be lower irreducibleif all itsk(n,k) submatrices

Received by the editors on 10 September 1996. Accepted for publication on 16 June 1998. Handling Editor: Daniel Hershkowitz.

yDepartamento de Matematica Aplicada, Universidad Politecnica de Valencia, 46020 Valencia, Spain ([email protected], [email protected], [email protected]). Research supported by Spanish DGICYT grant number PB94-1365-C03-02

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ELA

104 CristinaJordan,JuanR.Torregrosa,andAnaM.Urbano

[a ij]

k ; n

i=1;j=k +1 are nonzero for

k= 1;2;:::;n,1.

Given two nonincreasing sequences of nonnegative integers fk i

g p i=1 and

fq i

g r i=1,

we say thatfk i

g p

i=1 majorizes fq

i g

r

i=1, denoted fk

i g

p i=1

fq

i g

r i=1, if s

X i=1 k

i

s X i=1 q

i

; s= 1;2;:::;p; and p X i=1 k

i= r X

i=1 q

i :

Rodman and Shalom give in [9] the following completion problem,

Conjecture: LetAbe a lower irreducible partial upper triangularnnmatrix over F such that tr ace(A) = 0. Letn

1

n

2

n

p

1be a set of ppositive integers such that P

p i=1

n i =

n. There exists a nilpotent completion A c of

A whose Jordan form consists ofpblocks of sizesn

i n

i,

i= 1;2;:::;pif and only if

r(A k)

X i:n

i k

(n i

,k); k= 1;2;:::;n 1

:

(1)

Note that the right hand side of (1) is the rank of A k

c provided that A

c is a

nilpotent completion of A with the described Jordan form. The necessity of (1) is

therefore evident.

Rodman and Shalom prove the above conjecture whenr(A) = 1 (see [9], Theorem

3.2) or in general for matrices of size n4 (see [9], Theorem 3.3). In [5] we prove

that this conjecture is not true in general for matrices with minimal rank equal to three and for matrices of sizenn, n6.

In this paper we prove that this conjecture is true in two remaining cases: when

r(A) = 2, and for matrices of size 55.

LetA

0be the completion obtained by replacing the unspecied elements by zero.

We denote by J

q(

) the qq Jordan block with eigenvalue . The block diagonal

matrix with diagonal blocks A 1

;:::;A

t is denoted by A

1

A

t. Finally, given

a matrixA= [a ij] let

a

ij represent the corresponding nonzero entries of any similar

matrix obtained.

2. The main result.

We prove the conjecture of Rodman and Shalom for ma-trices with minimal rank equal to two.

Theorem 2.1. LetAbe annnlower irreducible partial upper triangular matrix such that tr ace(A) = 0 and r(A) = 2. Let n

1

n

2

n

p

1 be a set of p positive integers such that P

p i=1

n i=

n. If

r(A k)

X i:nik

(n i

,k); k= 1;2;:::;n 1

;

(2)

then there exists a nilpotent completion A c of

A whose Jordan form consists of p blocks of sizesn

i

n

i,

i= 1;2;:::;p.

Proof. By lower similarity we can transform the matrixA into a partial upper

triangular matrixA

1 which has only two nonzero rows. Since

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ELA

On the Rodman-Shalom conjecture 105

one of this rows is the rst one. We assume that the other one is the jth row. Then

the matrixA

1has the following structure:

A

Since the matrixA

1 is lower irreducible its nth column has a nonzero entry. We can

distinguish the following cases:

(a)a 1n

6

= 0.

By lower similarity we cancel the (j;n)th entry and all the entries in the rst row,

except the (1;1)th entry. Since the matrixA

1 is lower similar to the matrix A, its

minimalrank is equal to two, and then the jth row has a nonzero entry. Let us assume that:

(a1)a jt,

j<t<n, is the rst nonzero entry beginning from the right.

By lower similarity we can cancel all the entries in this row, except the elementa jj.

We obtain the matrix

A

completionA

2

0consists of

n,2 blocks of sizeq

If the sequencefn i

g p

i=1satises condition (2), then it majorizes the sequence fq

i g

n,2 i=1.

By performing the permutation

(1;n;2;3;:::;j,1;j;t;j+ 1;:::;t,1;t+ 1;:::;n,1)

of rows and columns in the matrixA

2and by applying the algorithm of [6] to the new

matrix (see Appendix, Part I) we obtain a completion of the matrixA

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Jordan form consists ofpblocks of sizen i,

i= 1;2;:::;p. This completion guarantees

the existence of one completion of the matrixA with the desired characteristics.

(a1.2) If a 11

6

= 0, we can transform, by lower similarity, the matrix A

2 into the

following matrix

A

This matrix satises that r(A

3) = 2 and

r(A 2

3) = 1. The Jordan form of the

completion A

3

0 consists of

n,2 blocks of size q

If the sequencefn i

g p

i=1satises condition (2), then it majorizes the sequence fq

i g

n,2 i=1.

By applying the method given in Appendix, Part II, we obtain a completion of the

matrixA

3, such that its Segre characteristic is fn

jj is the only nonzero entry in the jth row.

By lower similarity we can transform the matrixA 1 into

It is easily proved that the Jordan form of the completionA

20 consists of

n,2 blocks

i=1 satises

condition (2), then it majorizes the sequencefq i

g n,2

i=1. By performing the permutation

(1;j;n;2;3;:::;n,1) of rows and columns in the matrix A

2, and by applying the

algorithm of [6] (see Appendix, Part I) we obtain a completion of the matrixA 2such

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By lower similarity we transform the matrixA 1 into

wherea

1tis the rst nonzero entry in the rst row, starting from the right. Now, we

can distinguish the following subcases.

(b1)a 11=

a jj= 0.

If a

1j = 0 the Jordan form of the completion

A

20 consists of

n,2 blocks of size

n,2 = 1. Again, if the sequence

fn i

g p

i=1 satises

(2), then it majorizes the sequence fq i

g n,2

i=1 and we obtain the desired completion by

applying the algorithm of [6] to the matrix obtained performing the permutation (1;t;2;3;:::;j,1;j;n;j+ 1;:::;t,1;t+ 1;:::;n , 1)

of rows and columns in the matrixA

2(see Appendix, Part I).

Ifa 1j

6

= 0 we apply the method given in (a1.2), to the matrixA 2.

1j= 0 we transform, by lower similarity, the matrix A

We prove easily that the Jordan form of the completionA 3

0 consists of

n,3 blocks

n,3 = 1, and again that if the sequence fn

i g

p i=1

satises (2), then it majorizes the sequence fq i

g n,3

i=1. By performing the permutation

(1;j;t;n;2;3;:::;n,1) of rows and columns in the matrixA

3, and by applying the

algorithm of [6] to this new matrix (see Appendix, Part I) we achieve the desired completion.

Ifa 1j

6

= 0, by applying lower similarity to the matrixA

3 we obtain a new matrix as

in the case (b1) witha 1j

6

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The next theorem proves that the above conjecture is true for matrices of size

55.

Theorem 2.2. Let A be a lower irreducible partial upper triangular matrix of size55such that tr ace(A) = 0. Letn integers such thatP

p then there exists a nilpotent completion A

c of

A whose Jordan form consists of p blocks of sizesn

i

follows by applying Theorem 2.1 of [9]. Therefore, we can assume thatr(A) = 3.

If r(A 4)

>0 and r(A

5) = 0, the only possible Jordan form for some nilpotent

completion ofAisJ

5(0). The existence of this completion is guaranteed by Rodman

and Shalom in [9]. Ifr(A

3)

>0 andr(A

4) = 0, there are two possible Jordan forms for some nilpotent

completion ofA,J

The existence of the rst one is assured by the initial conditions about minimal rank forA

3) = 0. There are three possible Jordan

forms for some nilpotent completion of A, J 3(0)

Again, the last one exists by Theorem 2.1 of [9]. The rst structure exists as well because otherwise the minimal rank ofA

3 cannot be zero. Therefore, we can assure

thatr(A

2) = 1. Next we prove the existence of a nilpotent completion of the matrix Asuch that its Segre characteristic isf4;1g.

Let us assume thatA

c

0 is a completion of

Asuch that its Segre characteristic is f3;2g. Then, we distinguish the following cases:

(a)If the three rst rows ofA c

0 are linear independent, this matrix is lower similar

to

Since rank(A 2

In this case there exists a nonsingular matrixT

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wherea

3

assure thatJ Ac

0 = J

4(0) J

1(0), which is a contradiction. (a2)A

In this case there exists a nonsingular matrixT

1such that T

Next, consider the partial matrix

53= 0. The Jordan form of the matrix

obtained isJ 4(0)

J

1(0). This matrix allows us to achieve a completion of the initial

matrixA.

54= 0. The Jordan form of the matrix

obtained is J 4(0)

J

1(0). Again, this matrix allows us to get a completion of the

initial matrixA.

(b)If the independent rows of A

c0 are the rst one, the second one and the fourth

one, this matrix is lower similar to

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By performing the permutation (1;2;4;3;5) of rows and columns in Ac0 and by

ap-plying the above method, we obtain the desired completion.

(c) Finally, if the independent rows of Ac

0 are the rst one, the third one and the

fourth one, this matrix is lower similar to

Ac0

2 6 6 6 6 4

a11 a12 a13 a14 a15

0 0 0 0 0

c31 c32 a33 a34 a35

c41 c42 c43 a44 a45

0 0 0 0 0

3 7 7 7 7 5

:

Similarly, by performing the permutation (1;3;4;2;5) of rows and columns in Ac0 and

by applying the method given in (a) we achieve the desired completion.

3. App endix. Part I: Let Y be a matrix of size nm such that its m-diagonal

sums are s1;s2;:::sm. It is well known, see [10], that the matrix

Jm O

Y Jn

is lower similar to

Jm O

L Jn

; where

L =

2 6 6 6 6 6 4

0 0 0

0 0 ::: 0

... ... ...

0 0 ::: 0 s1 s2 ::: sm

3 7 7 7 7 7 5

:

The matrix L is said to be thelower concentrated formof the matrix Y . In the same way, if c1;c2;:::;cnare the n-diagonal sums the matrix

Jm O

Y Jn

is lower similar to

Jm O

C Jn

; where

C =

2 6 6 6 4

c1 0

0

c2 0 ::: 0

... ... ...

cn 0 ::: 0

3 7 7 7 5

:

The matrix C is calledleft concentrated form of the matrix Y . In general given a matrix

A =

2 6 6 6 6 6 4

Jn1 O

O O

Y21 Jn

2

O O

... ... ... ...

Yp,11 Yp,12

Jn

p,1 O

Yp1 Yp2

Ypp ,1 Jn

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by lower similarity we can transformA into the matrix

ii,1 is the left concentrated

form of the matrixY

ii,1, for

i= 2;3;:::;p. We call the matrix Ablock reduction to the rst columnof the matrixA.

The algorithm given in [6] is designed to work on a partial block upper triangular matrices, that is, matrices with the following structure

~

where the matrix X

ij has all its elements unknown, for

i > j, i = 2;3;:::;p and j= 1;2;:::;p,1.

Now consider the following partial block matrix

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ELA

and rank(Di0) = ni ,1.

-For i > j, i = 2;3;:::;p and j = 1;2;:::;p,1, the matrix B

ij has at least its rst

column formed by unspecied elements, being the rest of its elements unspecied or zero.

- For i < j, i = 1;2;:::;p,1 and j = 2;3;:::;p, the elements of block E

ij can be

unknown or equal to zero.

It is easy to see that the Segre characteristic of the completion M1 0 is

fn

1;n2;:::;np g.

Letfm j

g s

j=1be a sequence such that fn

i g

p i=1

fm j

g s

j=1. We obtain a completion of

the matrix M1 such that its Segre characteristic is fm

j g

s

j=1by applying the following

process:

(a)Replace by zeros all the unspecied elements of M

1, except the elements in the

rst column of blocks Bij, for i > j, i = 2;3;:::;p and j = 1;2;:::;p

,1. Therefore

we obtain:

M2=

2 6 6 6 6 6 4

D1

0 O O

O O

~B21 D20 O

O O

~B31 ~B32 D30

O O

... ... ... ... ...

~Bp1 ~Bp2 ~Bp3

~B

pp,1 Dp0 3 7 7 7 7 7 5

:

(b)By admissible similarity, the matrix M

2 is transformed into the matrix

M3= T

,1M 2T =

2 6 6 6 6 6 4

Jn

1 O O

O

S21 Jn

2 O

O

S31 S32 Jn 3

O

... ... ... ...

Sp1 Sp2 Sp3

J n

p 3 7 7 7 7 7 5

;

where the matrix Sij, for i > j, i = 2;3;:::;p and j = 1;2;:::;p

,1, has its rst

column formed by unknown elements, and the other entries equal to zero.

(c)Apply the Algorithm given in [6] to the matrix ~A in order to obtain a completion

~Ac such that its Segre characteristic is fm

j g

s j=1.

(d) Obtain the block reduction to the rst column of ~A

c and call ~Ac1 the matrix

obtained.

The matrix ~Ac1 is also a completion of M3. Since the matrix T is an admissible

similarity, T ~Ac1T

,1 is a completion of M

2 and therefore of M1 with the desired

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Example 3.1. Consider the partial block matrix

M

The Segre characteristic of the matrixM 1

1 such that its Segre characteristic is the sequence f5;4g.

Firstly we consider the following matrix

M

By using the admissible similarity

T =

we transformM

2 into the matrix

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ELA

By applying the Algorithm given in [6] to ~

where X

21 ;X

31 and X

32 have all their elements unspecied, we obtain ~

A

c, which

Segre characteristic is f5;4g. By doing its block reduction to the rst column we

obtain

The matrix

M

is a completion ofM

1 with the desired characteristic.

Part II: Consider the following lower irreducible partial upper triangular matrix,

A=

the completionA

0consists of

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If the sequence fn

i g

p

i=1 satises condition (2) then it majorizes the sequence fq

i g

n,2

i=1. We are going to obtain a completion

A

c of the matrix

A whose Segre

characteristic isfn i

g p i=1.

By performing the permutation (1;j;n;2;:::;j,1;j+ 1;:::;n,1) of rows and

columns inA, we obtain

A

n,3)3 whose rst column is formed by unspecied elements

andA

22is a partial upper triangular matrix whose known elements are equal to zero.

Next, consider the matrix

12 with a

1t = 0. The structure of

A

1 allows us to apply the

algorithm given in [6] and to obtain a completion

whose Segre characteristic isfn i

g p

i=1. We can distinguish two cases: (a)The rowt+ 1 is zero.

Then the matrix

A

is a completion ofA

1 similar to A

1c. (b)The rowt+ 1 of A

1c has an 1.

In this case, consider the matrix

A

completionA

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where

~

A 11

c= 2 4

0 a

1j a

1n

0 a

jn

0 0 0

3 5

; with =,a 1t

=a 1j

:

If the 1 of A

1c is in position (

t+ 1;h), with 3<hj+ 1, the matrixA 2c is

A 2c=

~

A 11

0 A

12 c A

21 c

A 22

c

;

where

A 12

c = 2 4

0 0 0 a

1t

0

0 0 0 0

0 0 0 0 0

3 5 ;

with=,a

1t =a

1j in position (2 ;h).

Finally, if the 1 is in position (t+ 1;h), withj+ 1<ht, by elementary

transfor-mations we obtain a completionA 2

c similar to A 1

c.

Example 3.2. Consider the partial upper triangular matrix

A= 2 6 6 6 6 6 6 6 6 6 6 4

0 0 1 0 0 1 0 1

0 0 0 0 0 0 0

0 0 0 0 0 1

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

3 7 7 7 7 7 7 7 7 7 7 5 :

We are going to nd a completion ofA such that its Segre characteristic isf7;1g.

First, by performing the permutation (1;3;8;2;4;5;6;7) we obtain the matrix

A 1=

2 6 6 6 6 6 6 6 6 6 6 4

0 1 1 0 0 0 1 0

0 1 0 0 0 0

0

0 0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

3 7 7 7 7 7 7 7 7 7 7 5 :

By applying the algorithm of [6] to the matrix A

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whose Segre characteristic isf7;1g

A 1c=

2 6 6 6 6 6 6 6 6 6 6 4

0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

3 7 7 7 7 7 7 7 7 7 7 5 :

Since there is an 1 in the row 7, consider the matrix

A 2=

2 6 6 6 6 6 6 6 6 6 6 4

0 1 1 0 0 0 1 0

0 1 0 0 0 0

0 0

1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

3 7 7 7 7 7 7 7 7 7 7 5 :

By applying the elementary transformationsC7,C2! C7,C6,C3 +C2! C6

andC5 +C3,C2!C5 (where Ciis the columniofA

2), and by its corresponding

transformations by rows, we obtain the matrix

A 2c =

2 6 6 6 6 6 6 6 6 6 6 4

0 1 1 0 0 0 1 0

0 0 1 ,1 0 0 0 0

0 0 0 1 ,1 0 0 0

1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

3 7 7 7 7 7 7 7 7 7 7 5 :

This matrix is a completion of A

2, similar to A

1c. By performing the permutation

(1;4;2;5;6;7;8;3) of rows and columns in the matrixA

2c we obtain a completion A

c

of the matrixA whose Segre characteristic isf7;1g. REFERENCES

[1] Joseph A. Ball, Israel Gohberg, Leiba Rodman and Tamir Shalom. On the eigenvaluesof matri-ces with given upper triangular part.IntegralEquationsandOperatorTheory, 13:488{497, 1990.

[2] Israel Gohberg and Sorin Rubinstein. A classication of upper equivalent matrices: the generic case.IntegralEquationsandOperatorTheory, 14:533{544, 1991.

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118 Cristina Jordan, Juan R. Torregrosa, and Ana M. Urbano

[4] Leonid Gurvits, Leiba Rodman and Tamir Shalom. Controllability of completion of partial upper triangular matrices.Mathematics of Control, Signals and Systems, 6:30{40, 1993. [5] Cristina Jordan, Juan R. Torregrosa and Ana M. Urbano. On the Jordan Form of Completions

of Partial Upper Triangular Matrices. Linear Algebra and its Applications, 254:241{250, 1997.

[6] Cristina Jordan, Juan R. Torregrosa and Ana M. Urbano. An Algorithm for Nilpotent Com-pletions of Partial Jordan Matrices.Linear Algebra and its Applications, 275/276:315{325, 1998.

[7] Mark Krupnik. Geometric multiplicities of completions of partial triangular matrices.Linear Algebra and its Applications, 220:215{227, 1995.

[8] Mark Krupnik and Leiba Rodman. Completions of partial Jordan and Hessenberg matrices.

Linear Algebra and its Applications, 212/213:267{287, 1994.

[9] Leiba Rodman and Tamir Shalom. Jordan form of completions of partial upper triangular matrices.Linear Algebra and its Applications, 168:221{249, 1992.