Journal of Mathematical Sciences, Vol. 157, No. 5, 2009
ON CIRCUIT INCLUSION SETS FOR THE SINGULAR VALUES OF A SQUARE MATRIX
UDC 512.643
∗
L. Yu. Kolotilina
The paper considers different circuit inclusion sets for the singular values of a square matrix. It is shown that in both the general and the structurally symmetric cases, the circuit inclusion sets are of theoretical interest only, because they coincide with the much simpler Ostrowski–Brauer type inclusion sets, which should be used in practice. Bibliography: 10 titles.
1. Introduction
In [1℄, Brualdi proved his famous theorem on the ir uit in lusion sets for the eigenvalues of a weakly irredu ible matrix. n×n ; n ≥ 2, is weakly irredu ible, then all its eigenvalues are Theorem 1.1 [1℄. If a matrix A = (aij ) ∈ C
ontained in the set
B (A) =
[
∈C(A): | |≥2
z∈C:
Y i∈
|z − aii | ≤
Y i∈
ri′ (A) :
(1:1)
In (1.1) and throughout the paper, the following notation is used: • hni = {1; : : : ; n}; n ∈ N; for a matrix A = (aij ) ∈ Cn×n , n P • ri′ (A) = |aij |; i = 1; : : : ; n, are the deleted absolute row sums of A; j =1;j 6=i
GA is the dire ted graph of A; • C(A) is the set of simple ir uits in GA ; re all that = (i1 ; : : : ; ik ; ik+1 ), where k ≥ 1 and ik+1 = i1 , is a simple ir uit in GA if for j = 1; : : : ; k the values ij are all distin t, and aij ij+1 6= 0; • for a ir uit = (i1 ; : : : ; ik ; ik+1 ) ∈ C(A), the set = {i1 ; : : : ; ik } is the support of , and | | = k is its •
length.
Re all that a matrix is said to be weakly irredu ible if every vertex of the graph GA belongs to the support of a ir uit ∈ C(A) of length | | ≥ 2. In [2℄, the following extension of Brualdi's theorem to the general ase was established. n×n ; n ≥ 1, are ontained in the Theorem 1.2 [2, Corollary 7.1℄. All the eigenvalues of a matrix A = (aij ) ∈ C set
B(A) =
[
∈C(A) | |≥2
z∈C:
Y i∈
|z − aii | ≤
Y i∈
ri′ (A)
[
aii : i ∈=
[
∈C(A): | |≥2
;
(1:2)
whi h is obtained by supplementing the set (1.1) with the union of the diagonal entries of A that are its irredu ible
omponents of order 1.
In a number of re ent publi ations, various in lusion sets for the singular values, analogous to the respe tive in lusion sets for the eigenvalues, were dis ussed (e.g., see [10, 7, 8, 6, 9, 5, 4, 3℄). In parti ular, the following
ir uit in lusion set, similar to the Brualdi set B(A), was suggested. n×n ; n ≥ 2. If Theorem 1.3 [6℄. Let A = (aij ) ∈ C in out Gout A (i) 6= ∅ and GA (i) ⊆ GA (i); i = 1; : : : ; n; ∗ St.Petersburg
(1:3)
Department of the Steklov Mathematical Institute, St.Petersburg, Russia, e-mail:
[email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 359, 2008, pp. 78–93. Original article submitted June 23, 2008. c 2009 Springer Science+Business Media, Inc. 1072-3374/09/1575-0715 715
then all the singular values of A are ontained in the set B(A) =
[
∈C(A): | |≥2
≥0:
Y i∈
| − |aii || ≤
Y i∈
s′i (A) :
(1:4)
In Theorem 1.3 and in what follows, we use the notation
′i (A) = ri′ (AT ); s′i (A) = max {ri′ (A); ′i (A)}; i = 1; : : : ; n; in Gout A (i) = {j ∈ hni \ {i} : aij 6= 0}; GA (i) = {j ∈ hni \ {i} : aji 6= 0}; i = 1; : : : ; n: Later the following Ky Fan type version of Theorem 1.3 was proved. n×n ; n ≥ 2, and let a nonnegative matrix G = (g ) ∈ Rn×n Theorem 1.4 [4℄. Let A = (aij ) ∈ C ij
ondition
gij ≥ max{|aij |; |aji |} for all i 6= j; i; j ∈ hni: If onditions (1.3) are ful lled, then all the singular values of A are ontained in the set KB(A) =
[
∈C(A): | |≥2
≥0:
Y i∈
| − |aii || ≤
i∈
((G) − gii ) ;
Y
(1:5) (1:6) satisfy the
(1:7) (1:8)
where (G) is the Perron root of the nonnegative matrix G, whose diagonal entries may be hosen arbitrarily.
As was pointed out in [6℄, the assumptions (1.3) of Theorems 1.3 and 1.4 on the properties of the graph GA imply that A is weakly irredu ible. However, as was demonstrated on an example in [4℄, Theorem 1.3 is not valid if A is only required to be weakly irredu ible. In this onne tion, in the latter paper it was onje tured that Theorem 1.4, on the ontrary, might hold for a weakly irredu ible matrix A, and it was also proved that under the assumptions of Theorem 1.4, the set (1.8) oin ides with the following simpler in lusion set: [ { ≥ 0 : | − |aii || | − |ajj || ≤ ((G) − gii ) ((G) − gjj )} : (1:9) i6=j : aij 6=0
As was demonstrated in [3℄, in the general ase the in lusion set (1.9) should be repla ed by the extended set [ [ { ≥ 0 : | − |aii || | − |ajj || ≤ ((G) − gii ) ((G) − gjj )} ∪ {|aii |}: (1:10) i6=j : aij 6=0
i∈hni: s′i (A)=0
In the present paper, we larify the situation with the ir uit in lusion sets for the singular values of a square matrix, using the general approa h developed in [3℄. The paper is organized as follows. In Se . 2, we prove an extension of Theorem 1.3 to the general ase (see Corollary 2.1), whi h is of theoreti al interest. On the other hand, we also show that the latter result proves to be of no pra ti al importan e be ause the ir uit set obtained a tually oin ides with the mu h simpler Ostrowski{Brauer type in lusion set (see Theorem 2.2). The nal result of this se tion (see Theorem 2.3) on erns the stru turally ( ombinatorially) symmetri matri es and, in parti ular, the matri es satisfying onditions (1.3). Here, it is pertinent to note that the ondition out Gin (1:11) A (i) ⊆ GA (i); i = 1; : : : ; n; o
urring in (1.3), amounts to the stru tural symmetry of the matrix A. Indeed, if aij 6= 0 for some i 6= j , i.e., out i ∈ Gin A (j ), then, by (1.11), i ∈ GA (j ), i.e., aji 6= 0, whi h shows that A is stru turally symmetri . Conversely, out if A is stru turally symmetri , then from j ∈ Gin A (i), i.e., from aji 6= 0 it follows that aij 6= 0, i.e., j ∈ GA (i), in out whi h shows that GA (i) ⊆ GA (i). Thus, onditions (1.3) split into two independent onditions, namely, the weak irredu ibility of A and its stru tural symmetry. In view of the above remark and Theorem 2.3, Theorem 1.3, as well as Corollary 2.1, is of no pra ti al interest. In Se . 3, the results of Se . 2 are generalized with the use of diagonal s aling matri es. Finally, in Se . 4, we present the Ky Fan type ounterparts of the in lusion sets onsidered in Se . 2 and prove the respe tive
oin iden e results. Also we disprove the above-mentioned onje ture on the validity of Theorem 1.4. 716
2. Brualdi type inclusion sets
The following simple result allows one to redu e the study of the singular spe trum (A) of a given matrix A to the singularity/nonsingularity problem for an auxiliary matrix. n×n ; n ≥ 1. Represent A as Lemma 2.1 [3℄. Let A = (aij ) ∈ C A = DA + B; where DA = diag (a11 ; : : : ; ann ): If ∈ (A), then the 2n × 2n matrix 2 In − |DA |2 0 DA B ∗ B ( ) A() = ( ij ) = − (2:1) 0 2 I − |D |2 B ∗ D∗ B n
is singular.
A
A
Note that the zero/nonzero pattern of A() is the same for all > 0. Therefore, the ir uit set C(A() ) also is the same for all > 0 and will be denoted simply by C. Observe that C ⊇ C(A(0) ). Using Lemma 2.1, we obtain the following basi result of this paper. Note that, in ontrast to Theorems 1.3 and 1.4, the theorem below involves the ir uit set C rather than C(A). n×n ; n ≥ 1, and let A() be de ned as in (2.1). Then the singular values of Theorem 2.1. Let A = (aij ) ∈ C A are ontained in the set
[
∈C: | |≥2
≥0:
Y i∈
( )
| ii | ≤
Y i∈
ri′ (A() )
∪
[
i∈hni: s′i (A)=0
{|aii |}:
(2:2)
Let ∈ (A); ≥ 0. We will show that if does not belong to the rst union in (2.2), then it ne essarily belongs to the se ond one, i.e., = |aii | for some i ∈ hni su h that s′i (A) = 0: (2:3) ( ) Indeed, by Lemma 2.1, the matrix A is singular, and therefore, by Theorem 1.2, the point z = 0 is ontained in the eigenvalue in lusion set B(A() ), de ned in (1.2). Thus, at least one of the following two onditions is ful lled: • either Y ( ) Y | ii | ≤ ri′ (A() ) for a ir uit ∈ C(A() ) of length | | ≥ 2 (2:4) Proof.
•
or
i∈
i∈
(kk) = 0 for some k ∈ h2ni su h that k ∈= for all ∈ C(A() ) of length | | ≥ 2: (2:5) ( ) Sin e C(A ) ⊆ C and, by our assumption on , ondition (2.4) annot be ful lled, ondition (2.5) must be ful lled, when e the equality (kk) = 0 holds for some k ∈ h2ni. This implies that = |aii |, where i ≡ k mod n and i ∈ hni, and onsequently (ii) = 0. Suppose s′i (A) 6= 0, i.e., either ri′ (A) 6= 0 or ′i (A) 6= 0. In the former
ase, there is an index j ∈ hni r {i} su h that aij 6= 0. Then, by (2.1), for any > 0 we have (i j)+n 6= 0 and (j +)n i 6= 0, implying that i ∈ , where = (i; j + n; i) ∈ C; | | = 2, whi h ontradi ts the assumption that does not belong to the rst union in (2.2). Therefore, ri′ (A) = 0. Similarly, if ′i (A) 6= 0, then we have aji 6= 0 for some j ∈ hni r {i}, implying that for any > 0, (j i)+n 6= 0 and (i+ )n j 6= 0. In this ase, i + n ∈ , where = (i + n; j; i + n) ∈ C; | | = 2. Sin e (i+)n i+n = 0, this again ontradi ts the assumption on . Thus, ′i (A) = 0, whi h proves (2.3) and ompletes the proof of the
theorem. Note that
ri′ (A() ) = |aii | ′i (A) + ri′ (A); ri+n (A ) = |aii |ri (A) + i (A); ′
and therefore
( )
′
i = 1; : : : ; n;
(2.6)
′
max {ri′ (A() ); ri′+n (A() )} ≤ (|aii | + )s′i (A); i = 1; : : : ; n; where s′i (A) is de ned in (1.5). In view of (2.7), Theorem 2.1 immediately implies the following result.
(2:7) 717
Corollary 2.1.
The singular values of an arbitrary matrix A = (aij ) ∈ Cn×n ; n ≥ 1, are ontained in the set
(A) = B where, for
[
∈C: | |≥2
1 ≤ i ≤ 2n, we set
≥0:
i=
Y Y ′ − |aii | ≤ si (A) ∪ i∈
i∈
[
i∈hni: s′i (A)=0
(2:8)
{|aii |};
i
if i ∈ hni; i − n if n < i < 2n:
(2.9)
In the general ase, the in lusion set B (A), de ned in (2.8), repla es the in lusion set B(A), de ned in Theorem 1.3, and, as we will see, B (A) oin ides with B(A) in the ase where onditions (1.3) are ful lled, i.e., the matrix A is weakly irredu ible and stru turally symmetri . The next result will be used in proving the oin iden e of di erent in lusion sets. n×n ; n ≥ 1, and arbitrary values t ≥ 0; i = 1; : : : ; n, the sets Lemma 2.2. For an arbitrary matrix A ∈ C i [
∈C: | |≥2
and
[
i6=j : aij 6=0
≥0:
Y Y − |aii | ≤ ti
i∈
(2:10)
i∈
{ ≥ 0 : | − |aii || | − |ajj || ≤ ti tj }
(2:11)
oin ide. Proof. First
we show that (2.10) is ontained in (2.11). To this end, let belong to the set (2.10). Then, for a
ir uit = (i1 ; : : : ; ik ; ik+1 = i1 ) ∈ C, where k ≥ 2, we have k k Y Y tij : − |aij ij | ≤
j =1
(2:12)
j =1
If the right-hand side in (2.12) is nonzero, then from (2.12) we derive 1=2 k − |ai i | − |ai i Y j j j +1 j +1 |
tij tij+1
j =1
implying that for a ertain j ∈ hki,
=
k − |ai i | Y j j
tij
j =1
≤ 1;
where ij 6= ij+1 and, for > 0; (ij i)j 6= 0: (2:13) On the other hand, if the right-hand side in (2.12) is zero, then, for some j ∈ hki, we have = |aij ij |; where, for > 0; (ij i)j 6= 0: In this ase, (2.13) holds trivially. Taking into a
ount (2.1) and (2.9), from the inequality (ij )ij 6= 0 we derive that • if ij > n and ij +1 > n, then aij ij 6= 0; • if ij > n and ij +1 ≤ n, then aij ij 6= 0; • if ij ≤ n and ij +1 > n, then aij ij 6= 0; • if ij ≤ n and ij +1 ≤ n, then aij ij 6= 0. Thus, in all the ases, either aij ij 6= 0 or aij ij 6= 0. Along with (2.13), this proves that belongs to (2.11), when e (2.10) is ontained in (2.11). Conversely, assume that ≥ 0 belongs to (2.11), i.e., | − |aii || | − |ajj || ≤ ti tj ; where either aij 6= 0 or aji 6= 0: In this ase, at least one of the ir uits (i; j + n; i) or (i + n; j; i + n) belongs to C, implying that belongs to (2.10). This shows that (2.11) is ontained in (2.10) and ompletes the proof of Theorem 2.2. Corollary 2.1 and Lemma 2.2 immediately imply the following result. − |aij ij | − |aij+1 ij+1 | ≤ tij tij+1 ;
+1
+1
+1
+1
+1
+1
+1
+1
718
+1
Theorem 2.2.
The singular values of an arbitrary matrix A ∈ Cn×n ; n ≥ 1, are ontained in the set [
i6=j : aij 6=0
≥ 0 : | − |aii || | − |ajj || ≤ s′i (A) s′j (A)
∪
[
i∈hni: s′i (A)=0
{|aii |};
(2:14)
(2.8). Note that in the ase where s′i (A) 6= 0 for all i = 1; : : : ; n, the in lusion set (2.14) obviously redu es to the set
whi h oin ides with
≥ 0 : | − |aii || | − |ajj || ≤ s′i (A) s′j (A) ;
[
i6=j : aij 6=0
(2:15)
whi h rst appeared in [7℄. Note also that the set (2.14) is a spe ial ase of a more general in lusion set for the singular spe trum of a matrix suggested in [3℄. The remaining results of this se tion on ern the stru turally symmetri matri es and, in parti ular, the matri es satisfying onditions (1.3). n×n ; n ≥ 1, be stru turally symmetri . Then, for arbitrary values t ≥ 0; i = Lemma 2.3. Let a matrix A ∈ C i 1; : : : ; n, the ir uit set [
∈C(A): | |≥2
≥0:
Y
i∈
| − |aii || ≤
(2:16)
ti
Y i∈
(2.11). Proof. The fa t that (2.16) is ontained in (2.11) is established using arguments similar to those used in the proof of Lemma 2.2. It remains to note that for a stru turally symmetri matrix A, (2.11) is trivially ontained in (2.16). Using Lemma 2.3, from Theorem 2.2 we obtain the following extension of Theorem 1.3. n×n ; n ≥ 1, are ontained Theorem 2.3. The singular values of a stru turally symmetri matrix A = (aij ) ∈ C
oin ides with
in the ir uit set
[
∈C(A): | |≥2
≥0:
Y
i∈
| − |aii || ≤
Y i∈
s′i (A)
∪
[
i∈hni: s′i (A)=0
{|aii |};
(2:17)
(2.14) and (2.8). Thus, for a stru turally symmetri matrix A, all the three sets (2.8), (2.14), and (2.17) are the same, and the simplest Ostrowski{Brauer type in lusion set (2.14) should be used in pra ti e. In parti ular, under onditions (1.3), implying that A is stru turally symmetri and weakly irredu ible, the Brualdi type set (1.4), o
urring in Theorem 1.3, oin ides with the set (2.15). whi h oin ides with
3. Generalized Brualdi type inclusion sets
In this se tion, we present some generalizations of the in lusion results obtained in Se . 2, involving diagonal s aling matri es. To this end, following [3℄, from the matrix A() we pass to the diagonally onjugated matrix 2 DA Dx−1 B ∗ Dx Dx−1 BDy ; In − |DA |2 0 ) = D−1 A() D A(x;y = − (3:1) x;y x;y Dy−1 B ∗ Dx DA∗ Dy−1 BDy 0 2 In − |DA |2 whi h also is singular whenever ∈ (A). Here, Dx;y = diag (x1 ; : : : ; xn ; y1 ; : : : ; yn ); Dx = diag (x1 ; : : : ; xn ); Dy = diag (y1 ; : : : ; yn ) are positive-de nite diagonal matri es asso iated with some positive ve tors x = (xi )ni=1 and y = (yi )ni=1 . ) Observing that the zero/nonzero stru ture of the matrix A(x;y
oin ides with that of A() and applying the ) arguments used in proving Theorem 2.1 to A(x;y , we arrive at the following generalization of the latter. 719
Let A = (aij ) ∈ Cn×n ; n ≥ 1. Then, for arbitrary positive ve tors x; y ∈ Rn , the singular values of A are ontained in the set Theorem 3.1.
[
∈C: | |≥2
≥0:
Y
i∈
where the matrix A() = ( (ij) ) is de ned in
Observing that
( )
| ii | ≤
Y i∈
) ri′ (A(x;y )
(2.1).
∪
[
i∈hni: s′i (A)=0
(3:2)
{|aii |};
) ri′ (A(x;y ) = |aii |ri′ (Dx−1 A∗ Dx ) + ri′ (Dx−1 ADy ); i = 1; : : : ; n; ) ) = |a |r′ (D−1 AD ) + r′ (D−1 A∗ D ); i = 1; : : : ; n; ri′+n (A(x;y ii i y x y i y
and
) ); r′ (A() ) max ri′ (A(x;y i+n x;y
n
o
≤ (|aii | + )Si (A; x; y );
i = 1; : : : ; n;
where we set Si (A; x; y) = max ri′ (Dx−1 A∗ Dx ); ri′ (Dx−1 ADy ); ri′ (Dy−1 ADy ); ri′ (Dy−1 A∗ Dx ) ; i = 1; : : : ; n; (3:3) we immediately arrive at the following impli ation of Theorem 3.1, generalizing Corollary 2.1. n×n ; n ≥ 1. Then, for arbitrary positive ve tors x; y ∈ Rn , the singular values Corollary 3.1. Let A = (aij ) ∈ C of A are ontained in the set
[
∈C: | |≥2
≥0:
Y Y − |aii | ≤ Si (A; x; y) ∪ i∈
i∈
[
i∈hni: s′i (A)=0
(3:4)
{|aii |}:
Using Lemma 2.2, from Corollary 3.1 we obtain the following generalization of Theorem 2.2. n×n ; n ≥ 1. Then, for arbitrary positive ve tors x; y ∈ Rn , the singular values Theorem 3.2. Let A = (aij ) ∈ C of A are ontained in the set [
i6=j : aij 6=0
{ ≥ 0 : | − |aii || | − |ajj || ≤ Si (A; x; y ) Sj (A; x; y )}
∪
[
i∈hni: si (A)=0
{|aii |};
(3:5)
(3.4). Note that the in lusion set (3.5) is the spe ial ase orresponding to = 1=2 of a more general set obtained in [3℄. In the stru turally symmetri ase, Theorem 3.2 and Lemma 2.3 yield the result below. n×n ; n ≥ 1, be stru turally symmetri . Then, for arbitrary positive Theorem 3.3. Let a matrix A = (aij ) ∈ C whi h oin ides with the ir uit set
ve tors x; y ∈ Rn , the singular values of A are ontained in the ir uit set [
∈C(A): | |≥2
≥0:
Y i∈
| − |aii || ≤
Y i∈
Si (A; x; y)
∪
[
i∈hni: s′i (A)=0
{|aii |};
(3:6)
(3.5) and (3.4). We on lude this se tion by mentioning that from Theorem 3.2 it immediately follows that the in lusion set (3.5) an be repla ed by any set of the form \ [ [ { ≥ 0 : | − |aii || | − |ajj || ≤ Si (A; x; y ) Sj (A; x; y )} ∪ {|aii |}; (3:7) whi h oin ides with
x;y i6=j:
aij 6=0
i∈hni: si (A)=0
where the interse tion is taken over some (or all) positive ve tors x and y. This is an additional reason for )
onsidering the generalized in lusion sets arising from repla ing A() by A(x;y . Yet another reason is related to the possibility of deriving Ky Fan type in lusion results for the singular values in a very simple way. This issue will be addressed in the next se tion. 720
4. Ky Fan type inclusion sets
In this se tion, motivated by Theorem 1.4 and the oin iden e of the in lusion sets (1.8) and (1.9) under assumptions (1.3), whi h was established in [4℄, we onsider the Ky Fan type ounterparts of the in lusion sets (2.8), (2.14), and (2.17). To this end, let G = (gij ) ∈ Rn×n be a nonnegative matrix satisfying the (entrywise) inequalities G ≥ |A − DA | and G ≥ |A − DA |T ;
(4:1)
Gv = (G)v;
(4:2)
and let
where (G) is the Perron root of G, and v = (vi ) is its right Perron ve tor. If the ve tor v is positive, then the matrix Dv−1 is well de ned, and, in view of (3.3), (4.1), and (4.2), we have Si (A; v; v) = max ri′ (Dv−1 A∗ Dv ); ri′ (Dv−1 ADv )
≤ ri′ (Dv−1 GDv ) = (G) − gii ;
i = 1; : : : ; n:
(4:3)
By applying Corollary 3.1 with x = y = v and using (4.3), we immediately obtain (A) ⊆
[
∈C: | |≥2
≥0:
Y Y − |aii | ≤ ((G) − gii ) ∪ i∈
i∈
[
i∈hni: s′i (A)=0
{|aii |}:
(4:4)
In the general ase, where the Perron ve tor v of the matrix G is nonnegative, the validity of (4.4) is established using the ontinuity argument (by onsidering the positive matrix G" that is obtained from G by repla ing its zero entries with " > 0 and by passing to the limit as " → 0). Thus, we have proved the following Ky Fan type in lusion result. n×n ; n ≥ 1, and an arbitrary nonnegative matrix Theorem 4.1. For an arbitrary matrix A = (aij ) ∈ C n ×n G = (gij ) ∈ R satisfying onditions (4.1), the singular values of A are ontained in the set [
∈C: | |≥2
≥0:
Y Y − |aii | ≤ ((G) − gii ) ∪ i∈
i∈
[
i∈hni: s′i (A)=0
(4:5)
{|aii |}:
Applying Lemma 2.2, we arrive at the following Ky Fan type ounterpart of Theorem 2.2. Theorem 4.2. Under the assumptions of Theorem 4.1, the singular values of A are ontained in the set [
i6=j : aij 6=0
{ ≥ 0 : | − |aii || | − |ajj || ≤ ((G) − gii ) ((G) − gjj )}
∪
[
i∈hni: si (A)=0
{|aii |};
(4:6)
(4.5). In the stru turally symmetri ase, Lemma 2.3 and Theorem 4.2 imply the nal result of this se tion, extending Theorem 1.4. n×n ; n ≥ 1, be stru turally symmetri . Then, for an arbitrary matrix Theorem 4.3. Let a matrix A = (aij ) ∈ C n ×n G = (gij ) ∈ R satisfying onditions (4.1), the singular values of A are ontained in the set whi h oin ides with
[
∈C(A): | |≥2
whi h oin ides with the sets
≥0:
Y i∈
| − |aii || ≤
(4.5) and (4.6).
i∈
((G) − gii )
Y
∪
[
i∈hni: s′i (A)=0
{|aii |};
(4:7)
721
Note that for a matrix A satisfying onditions (1.3), whi h is weakly irredu ible and stru turally symmetri , Theorem 4.3 trivially implies the oin iden e of the sets (1.8) and (1.9), proved in [4℄ and mentioned in the Introdu tion. In on lusion, we show that for (weakly) irredu ible matri es that are not stru turally symmetri , Theorem 1.4 is in general not valid, whi h disproves the onje ture advan ed in [4℄. Indeed, onsider the irredu ible matrix 0 0:1 0 A = 0 2 1; 0:1 0 0
whi h di ers from the example onsidered in [4℄ in the se ond diagonal entry. We have 0:01 0:2 0 AA∗ = 0:2 5 0 0 0 0:01
and, as is not diÆ ult to as ertain, 1 (A) =
As in [4℄, set
i1=2
5:01 + 5:012 − 0:04 =2
h
p
= 2:2378 : : : :
(4:8)
0 0:1 0:1 G = 0:1 0 1 : 0:1 1 0 Then G satis es the assumptions of Theorem 1.4, and
(G) = 1:0196 : : : :
(4:9)
It remains to observe that, in view of (4.8) and (4.9), we have 1 (A)2 (1 (A) − 2) > (G)3 ;
whi h shows that 1 (A) is not ontained in the set (1.8). Translated by L. Yu. Kolotilina. REFERENCES
1. R. A. Brualdi, \Matri es, eigenvalues, and dire ted graphs," Linear Multilinear Algebra, 11, 143{165 (1982). 2. L. Yu. Kolotilina, \The singularity/nonsingularity problem for matri es satisfying diagonal dominan e onditions in terms of dire ted graphs," Zap. Nau hn. Semin. POMI, 309, 40{83 (2004). 3. L. Yu. Kolotilina, \In lusion sets for the singular values of a square matrix," Zap. Nau hn. Semin. POMI, 359, 52{77 (2008). 4. H.-B. Li, T.-Z. Huang, and H. Li, \In lusion sets for singular values," Linear Algebra Appl., 428, 2220{2235 (2008). 5. H.-B. Li, T.-Z. Huang, H. Li, and S.-Q. Shen, \Optimal Gers hgorin-type in lusion intervals of singular values," Numer. Linear Algebra Appl., 14, 115{128 (2007). 6. J. S. Li, K. Yang, and Q.-X. Wang, \Digraphs and in lusion intervals of Brualdi-type for singular values," A ta Math. Appl. Sin., 18, 471{476 (2002). 7 L. Li, \The undire ted graph and estimates of matrix singular values," Linear Algebra Appl., 285, 181{188 (1998). 8. L. L. Li, \Estimation of matrix singular values," Comput. Math. Appl., 37, 9{15 (1999). 9. W. Li and Q. Chang, \In lusion intervals of singular values and appli ations," Comput. Math. Appl., 45, 1637{1646 (2003). 10. L. Qi, \Some simple estimates for singular values of a matrix," Linear Algebra Appl., 56, 105{119 (1984). 722
