Joint bounds for the Perron roots of nonnegative matrices with

Journal of Mathematical Sciences, Vol. 141, No. 6, 2007

JOINT BOUNDS FOR THE PERRON ROOTS OF NONNEGATIVE MATRICES WITH APPLICATIONS Yu. A. Al’pin,∗ L. Yu. Kolotilina,† and N. N. Korneeva∗

UDC 512.643

Given a finite set {Ax }x∈X of nonnegative matrices, we derive joint upper and lower bounds for the row sums of the matrices D −1 A(x) D, x ∈ X, where D is a specially chosen nonsingular diagonal matrix. These bounds, depending only on the sparsity patterns of the matrices A(x) and their row sums, are used to obtain joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, bounds for the joint and lower spectral radii of a matrix set, and conditions sufficient for all convex combinations of given matrices to be Schur stable. Bibliography: 20 titles.

1. Introduction Let A = (aij ) ∈ Cn×n , n ≥ 1. Introduce into consideration the absolute row sums ri (A) =

n


|aij |,

i = 1, . . . , n,

j=1

and denote n = {1, . . . , n}. As is well known, A∞ = max ri(A) i∈n

is a multiplicative matrix norm, and, for a nonnegative matrix A, its spectral radius ρ(A), which is referred to as the Perron root of A, satisfies the two-sided Frobenius bounds min ri (A) ≤ ρ(A) ≤ max ri (A).

i∈n

(1.1)

i∈n

Since for any nonsingular diagonal matrix D we obviously have ρ(D−1 AD) = ρ(A), the following problem naturally arises from (1.1): find diagonal matrices D1 and D2 with positive diagonal entries, different from the identity matrix In , such that max ri(D1−1 AD1 ) ≤ max ri (A)

(1.2)

min ri(D2−1 AD2 ) ≥ min ri(A).

(1.3)

i∈n

and

i∈n

i∈n

i∈n

As is not difficult to ascertain, in the irreducible case there always exists a diagonal matrix D with positive diagonal entries for which (1.4) max ri(D−1 AD) = min ri (D−1 AD) = ρ(A). i∈n

i∈n

Indeed, by the Perron–Frobenius theorem, the Perron vector v = (vi ) of the matrix A, which is determined uniquely up to a positive multiplier, is positive, and from the equality Av = ρ(A)v one can readily see that equalities (1.4) hold for D = Dv = diag (v1 , . . . , vn ). Obviously, the choice D = Dv is theoretically optimal. But in practice it requires that the Perron vector be available. Therefore, it is of interest to find a less expensive method for determining nontrivial diagonal matrices D1 and D2 that satisfy conditions (1.2) and (1.3). This problem is closely related to the problem of bounding the Perron root of a nonnegative matrix but is more general. Implicitly it was considered in [1], where the following result was established (also see [15]). ∗ Kazan’

State University, Kazan’, Russia, e-mail: [email protected].

† St.Petersburg

Department of the Steklov Mathematical Institute, St.Petersburg, Russia, e-mail: [email protected].

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 30–56. Original article submitted May 29, 2006. 1586

c 2007 Springer Science+Business Media, Inc. 1072-3374/07/1416-1586 

Theorem 1.1. Let A ∈ Cn×n , n ≥ 1, be a nonnegative matrix free of zero rows. Set

C(A) = max

 

γ∈C(A)  i∈¯ γ

1/|γ|  ri (A) , 

c(A) = min

 

γ∈C(A)  i∈¯ γ

1/|γ|  ri(A) 

(1.5)

and define the diagonal matrices D = diag (d1 , . . . , dn ) and G = diag (g1 , . . . , gn ) by the relations di = Then

max (i=i1 ,...,ik ,ik+1 )∈P(A)

k 

[rij (A)/C(A)],

j=1

gi =

min (i=i1 ,...,ik ,ik+1 )∈P(A)

k 

[rij (A)/c(A)],

i = 1, . . . , n.

min ri(A) ≤ c(A) ≤ min ri (G−1 AG) ≤ ρ(A) ≤ max ri (D−1 AD) ≤ C(A) ≤ max ri(A).

i∈n

i∈n

(1.6)

j=1

i∈n

i∈n

(1.7)

Furthermore, if A is irreducible, then either c(A) = ρ(A) = C(A) or c(A) < ρ(A) < C(A). In Theorem 1.1 and below, we use the following notation. For a matrix A = (aij ) ∈ Cn×n , n ≥ 1, GA = (n, EA ) is the directed graph of A with vertex set n = {1, . . . , n} and arc set EA = {(i, j) : i, j ∈ n

and aij = 0}.

C(A) denotes the set of all (simple) circuits in GA . Recall that a circuit of length k in GA is an ordered sequence γ = (i1 , . . . , ik , ik+1 ), where all i1 , . . . , ik ∈ n are distinct, ik+1 = i1 , and for each j = 1, . . . , k there is an arc in GA going from ij to ij+1 . The set {i1 , . . . , ik } is called the support of γ and is denoted by γ¯. The length of the circuit γ is denoted by |γ|; |γ| = k. The set P(A) of paths in GA is defined as follows: P(A) = {(i1 , . . . , ik , ik+1 ) : ai1 i2 · · · aik ik+1 = 0, k ≥ 0}, and by SPi (A), i ∈ n, we denote the set of all simple paths in GA that outgo from vertex i. It should be mentioned that from the proof presented in [1] it follows that the maximum and minimum in (1.6) can actually be taken over SPi (A), which noticeably simplifies the construction of the diagonal matrices D and G. As is readily seen, in the case where all the diagonal entries of A are nonzero, the rightmost and leftmost inequalities in (1.7) hold with equality. For this reason, Theorem 1.1 may actually improve the Frobenius bounds (1.1) only for matrices having sufficiently many zero diagonal entries. In this paper, we consider a still more general problem, namely: given a finite set of n × n matrices Σ = {A(x) }x∈X , find nonsingular diagonal matrices D1 and D2 , different from the identity matrix In , such that max max ri(D1−1 A(x)D1 ) ≤ max max ri(A(x) )

(1.8)

min min ri (D2−1 A(x) D2 ) ≥ min min ri (A(x) ).

(1.9)

x∈X i∈n

and

x∈X i∈n

x∈X i∈n

x∈X i∈n

Of course, we aim at minimizing the left-hand side of (1.8) and at maximizing the left-hand side of (1.9). The paper is organized as follows. Our main results, concerning the solution of problems (1.8) and (1.9), are established in Sec. 2. Section 3 considers some applications of the results of Sec. 2. Specifically, joint two-sided bounds for the Perron roots of given nonnegative matrices, joint upper bounds for the spectral radii of given complex matrices, conditions sufficient for all convex combinations of given matrices to be Schur stable, and bounds for the joint and lower spectral radii of a set of matrices are derived. 1587

2. Main results In this section, given a finite set Σ = {A(x) }x∈X of nonnegative matrices of order n ≥ 1, we derive joint upper and lower bounds for the row sums ri(D−1 A(x) D), i ∈ n, x ∈ X, for some special choices of the nonsingular diagonal matrix D. In what follows, we will use the auxiliary matrix ¯ = (¯ A aij ),

(x)

a ¯ij = max aij .

(2.1)

x∈X

The simplest way to derive joint upper bounds for the row sums of the set D−1 ΣD is to pass to the matrix ¯ from above, for instance, by applying Theorem 1.1 or A¯ defined in (2.1) and to bound the row sums of D−1 AD ¯ (Lower bounds can be obtained in a similar way if, in (2.1), max another bound in terms of the row sums of A. x∈X ¯ is replaced by min.) Joint bounds obtained in this way will obviously depend on the row sums of the matrix A. x∈X

However, as we will see, it is possible to obtain joint bounds for the row sums of the matrices D−1 A(x) D that depend only on the sparsity patterns and row sums of the given matrices A(x) . The first result of this kind is as follows. (x)

Theorem 2.1. Let Σ = {A(x) }x∈X be a finite set of nonnegative matrices A(x) = (aij ) of order n ≥ 1 such that the matrix A¯ is free of zero rows, and let 0 ≤ α ≤ 1. Then, for where di = max ri (A(x) )1−α ,

D = diag (d1 , . . . , dn ), we have

α

1−α  max ri (A(x) ) max rj (A(x) ) ≤ max max ri (A(x) ).



max max ri (D−1 A(x) D) ≤ x∈X i∈n

max

i,j: ¯ aij >0

i = 1, . . . , n,

x∈X

x∈X

(x)

x: aij >0

x∈X i∈n

(2.2)

(2.3)

Similarly, if all the matrices A(x) , x ∈ X, are free of zero rows, then, for D = diag (d1 , . . . , dn ), we have



min min ri (D−1 A(x) D) ≥

x∈X i∈n

min

i,j: a ¯ij >0

where

di = min ri (A(x) )1−α ,

i = 1, . . . , n,

x∈X

α

1−α  min ri(A(x) ) min rj (A(x) ) ≥ min min ri(A(x) ). x∈X

(x)

x: aij >0

x∈X i∈n

(2.4)

(2.5)

Proof. First we note that the assumption that A¯ has no zero rows implies that all the values di defined in (2.2) are positive, whereas the assumption that all the matrices A(x) , x ∈ X, are free of zero rows implies that all the values di defined in (2.4) are positive. In order to prove the left inequality in (2.3), we derive ri(D

−1

(x)

A

(x)

D) = max ri(A x∈X

−(1−α) )




(x)

aij dj

(x)

j: aij >0



−(1−α) ≤ max ri (A(x) ) ri(A(x) ) max dj x∈X

(x)

j: aij >0

  (x) α (x) 1−α ≤ max ri(A ) max rj (A ) , The relations obtained imply the inequalities max ri(D x∈X

1588

−1

(x)

A

D) ≤ max

j: ¯ aij >0

x ∈ X.

i = 1, . . . , n,

x∈X

(x)

j: aij >0

 (x) α

max ri (A (x)

x: aij >0

)

(x) 1−α

max rj (A x∈X

)

,

i = 1, . . . , n,

from which the left inequality in (2.3) readily follows. The right inequality in (2.3) holds trivially. Inequalities (2.5) are derived in a similar way.
Note that in the case where all matrices A(x) , x ∈ X, have the same sparsity pattern, bounds (2.3) and (2.5) take the simpler forms

α

1−α  −1 (x) (x) (x) max max ri (D A D) ≤ max max ri (A ) max rj (A ) (2.6) x∈X i∈n

i,j: ¯ aij >0

min min ri (D

x∈X i∈n

x∈X



and −1

x∈X

(x)

A

D) ≥

(x)

min

i,j: ¯ aij >0

min ri(A

x∈X

α

1−α  (x) min rj (A ) . ) x∈X

(2.7)

In particular, for a single nonnegative matrix, Theorem 2.1 yields the following new result. Corollary 2.1. Let A be a nonnegative matrix of order n ≥ 1 free of zero rows, let 0 ≤ α ≤ 1, and let D = diag (d1 , . . . , dn ),

where

di = ri(A)1−α ,

i = 1, . . . , n.

(2.8)

Then min ri (A) ≤

i∈n

≤

min

i,j: aij >0



 ri(A)α rj (A)1−α ≤ ri (D−1 AD) (2.9)

  max ri (A)α rj (A)1−α ≤ max ri(A),

i,j: aij >0

i = 1, . . . , n.

i∈n

It should be pointed out that for a single matrix, both upper and lower bounds are obtained for the same diagonal transformation matrix D, which, in general, is not the case for a set of matrices. Note that in Corollary 2.1, only the row sums ri(A), i = 1, . . . , n, and the sparsity pattern of A are used, and consequently bounds (2.9) are shared by all the nonnegative matrices having the same row sums and sparsity pattern as A. Our next result provides an improvement of bounds (2.3) and (2.5) at the expense of a more sophisticated choice of the conjugating diagonal matrices D. Simultaneously, it extends Theorem 1.1 to finite sets of nonnegative matrices. (x)

Theorem 2.2. Let Σ = {A(x)}x∈X be a finite set of nonnegative matrices A(x) = (aij ) of order n ≥ 1. Set

C(Σ) =

c(Σ) =

 s 

max

max

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

 s 

min

min

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

1/s  rij (A(x) ) ,  1/s  rij (A(x) ) . 

(2.10)

If the matrix A¯ defined in (2.1) is free of zero rows, then for D = diag (d1 , . . . , dn ), where di =

r 

max

¯ (i1 ,...,ir ,ir+1 )∈SPi (A)

we have the upper bound

max (x)

  rij (A(x) )/C(Σ) ,

i = 1, . . . , n,

(2.11)

j=1 x: aij ij+1 >0

max max ri (D−1 A(x) D) ≤ C(Σ),

(2.12)

x∈X i∈n

and if all the matrices A(x) , x ∈ X, are free of zero rows, then for D = diag (d1 , . . . , dn ), where di =

min

¯ (i1 ,...,ir ,ir+1 )∈SPi (A)

r 

 min

(x)

 rij (A(x) )/c(Σ) ,

i = 1, . . . , n,

(2.13)

j=1 x: aij ij+1 >0

1589

we have the lower bound

min min ri(D−1 A(x) D) ≥ c(Σ).

(2.14)

x∈X i∈n

  Proof. In order to prove the bound (2.12), consider the set of scaled matrices Σ = C(Σ)−1 A(x) . As is easy to ascertain, the set Σ satisfies the equality (2.15) C(Σ ) = 1. From the definition of the matrix D it follows that for every x ∈ X and arbitrary i ∈ n, we have


C(Σ)−1 aij dj ≤ C(Σ)−1 ri (A(x) )

max dj = C(Σ)−1 ri (A(x) ) dj0 ,

(x)

j

(x)

j: aij >0

where j0 is a vertex for which dj0 = max dj . Then, taking into account (2.15), we derive the inequalities (x)

j: aij >0




C(Σ)−1 aij dj ≤ di,

i = 1, . . . , n,

x ∈ X,

ri (D−1 A(x) D) ≤ C(Σ),

i = 1, . . . , n,

x ∈ X.

(x)

j

immediately implying that

This proves the bound (2.12). The lower bound (2.14) is proved similarly, but as the scaling factor one should take c(Σ).
Note that in the case where the matrix set consists of a single matrix A, the results of Theorem 2.2 reduce to those of Theorem 1.1. Remark 2.1. As is readily seen, the bounds (2.12) and (2.14) are at least as good as the bounds (2.3) and (2.5). This follows from the obvious relations

min

1≤i≤s



aα i

a1−α i+1



 ≤ (a1 · · · as )

1/s

=

s 

1/s aα i

a1−α i+1

i=1

  1−α ≤ max aα i ai+1 , 1≤i≤s

where we set as+1 = a1 , which hold for arbitrary ai > 0, i = 1, . . . , s, and 0 ≤ α ≤ 1. However, the bounds (2.3) and (2.5) are much easier to compute than the bounds (2.12) and (2.14). Note also that if all the matrices (x) (x) A(x) , x ∈ X, have the same sparsity pattern and the latter is symmetric, i.e., aij > 0 ⇐⇒ aji > 0, i = j, x ∈ X, then, for any α ∈ [0, 1], the bounds (2.12) and (2.14) coincide with the bounds (2.6) and (2.7), respectively. (x) ¯ This stems from the fact that if aij > 0, then (i, j, i) ∈ C(A(x) ) = C(A). 3. Applications In this section, some applications of Theorems 2.1 and 2.2 are presented. 3.1. Joint bounds for the Perron roots In view of the Frobenius bounds (1.1), Theorem 2.1 immediately implies the joint two-sided bounds for the Perron roots of nonnegative matrices from a finite set presented in the following theorem. (x)

Theorem 3.1. Let Σ = {A(x)}x∈X be a finite set of nonnegative matrices A(x) = (aij ) of order n ≥ 1 free of zero rows and let 0 ≤ α ≤ 1. Let the matrix A¯ = (¯ aij ) be as defined in (2.1). Then

(x)

min

min ri (A

i,j: a ¯ij >0

(x)



≤ 1590

max

i,j: ¯ aij >0

x: aij >0

α

1−α  (x) ) min rj (A ) ≤ ρ(A(x) ) x∈X

α

1−α  (x) (x) max ri (A ) max rj (A ) , (x)

x: aij >0

x∈X

(3.1) x ∈ X.

Furthermore, if a matrix A(x) , x ∈ X, is irreducible, then the equality

α ρ(A(x) ) =

max

i,j: ¯ aij >0

1−α  max rj (A(x) )

max ri (A(x) )

x∈X

(x)

x: aij >0

(3.2)

holds if and only if the equality

(x) α

ri (A

(x) 1−α

) rj (A

)

=

(x)

max ri(A

max

i,j: a ¯ij >0

α

1−α  (x) max rj (A ) ) x∈X

(x)

x: aij >0

(3.3)

(x)

holds for all i, j ∈ n such that aij > 0. Similarly, for an irreducible matrix A(x) , x ∈ X, the equality

α

1−α  (x) (x) (x) min ri(A ) min rj (A ) ρ(A ) = min i,j: a ¯ij >0

x∈X

(x)

x: aij >0

(3.4)

holds if and only if

(x) α

ri(A

(x) 1−α

) rj (A

)

=

(x)

min ri (A

min

i,j: a ¯ij >0

(x)

x: aij >0

α

1−α  (x) ) min rj (A ) x∈X

(3.5)

(x)

for all i, j ∈ n such that aij > 0. Proof. Only the assertions concerning equalities (3.2) and (3.4) must be proved. Assume that A(x) is irreducible and that equality (3.2) is valid. Then we have   ρ(A(x) ) ≥ max ri(A(x) )α rj (A(x) )1−α . (3.6) (x)

i,j: aij >0

On the other hand, by the Frobenius bounds (1.1) and Corollary 2.1 (or Corollary 5.1 in [14]), we have the opposite inequality   ρ(A(x) ) ≤ max ri(A(x) )α rj (A(x) )1−α , (x)

i,j: aij >0

which, in conjunction with (3.6), yields the equality ρ(A(x) ) =

 max (x)

 ri(A(x) )α rj (A(x) )1−α .

(3.7)

i,j: aij >0

But, by Corollary 5.1 in [14], equality (3.7) holds true for the irreducible matrix A(x) if and only if ri (A(x) )α rj (A(x) )1−α = ρ(A(x) ) for all i, j

such that

(x)

aij > 0.

The latter equalities, together with (3.2), prove (3.3). (x) Conversely, if equality (3.3) holds for i, j such that aij > 0, then equality (3.2) immediately stems from Corollary 2.1 and the Frobenius bounds (1.1). The case of equality (3.4) is considered similarly.
Remark 3.1. Since the upper bound of Theorem 2.1 holds under the weaker assumption that the matrix A¯ is free of zero rows, the upper bound in (3.1) obviously holds under the same assumption. Furthermore, as is easy to ascertain, it actually holds for an arbitrary set of nonnegative matrices such that A¯ has a principal submatrix ¯ = 0, implying that ρ(A(x) ) = 0, x ∈ X. free of zero rows. If no such submatrix exists, then, obviously, ρ(A) As to the lower bound in (3.1), it can be extended to an arbitrary set of nonnegative matrices such that for a (x) nonempty subset S ⊆ n, all the principal submatrices A(x) [S] = (aij )i,j∈S are free of zero rows. To this end, in view of the monotonicity property of the Perron root with respect to principal submatrices, suffice it to take the minimum over all i, j ∈ S such that a ¯ij > 0. For a single nonnegative matrix A without zero rows, Theorem 3.1 and Corollary 5.1 in [14] imply the following more precise result, originally obtained in [14]. 1591

Corollary 3.1. Let A = (aij ) be a nonnegative matrix of order n ≥ 1 free of zero rows. Then     ri(A)α rj (A)1−α ≤ ρ(A) ≤ max ri (A)α rj (A)1−α . min i,j: aij >0

i,j: aij >0

(3.8)

Furthermore, if A is irreducible, then both inequalities in (3.8) are equalities if and only if • either ri(A) = ρ(A), i = 1, . . . , n, • or the following conditions are simultaneously fulfilled: (i) α = 1/2; (ii) n = S1 ∪ S2 , where, for a certain ξ > 1, S2 = {i ∈ n : ri(A) = ξ −1 ρ(A)};

S1 = {i ∈ n : ri(A) = ξρ(A)}, (iii) both principal submatrices A[S1 ] and A[S2 ] are zero; otherwise both inequalities in (3.8) hold strictly.

The joint two-sided bounds for the Perron roots presented below trivially follow from Theorem 2.2 and generalize the circuit bounds of Theorem 1.1 to sets of nonnegative matrices. (x)

Theorem 3.2. Let Σ = {A(x) }x∈X be a finite set of nonnegative matrices A(x) = (aij ) of order n ≥ 1. If the matrices A(x) , x ∈ X, are free of zero rows, then c(Σ) ≡

 s 

min

min

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

≤ C(Σ) ≡

max

 s 

max

1/s  rij (A(x) ) ≤ ρ(A(x) ) 

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

1/s  rij (A(x) ) , 

(3.9) x ∈ X,

where A¯ is defined in accordance with (2.1). Furthermore, if a matrix A(x) , x ∈ X, is irreducible, then the equality ρ(A(x) ) = C(Σ) (3.10) holds if and only if

 



1/|γ| ri (A(x) )

= C(Σ) for every

γ ∈ C(A(x) ),

(3.11)

i∈¯ γ

and, similarly, the equality ρ(A(x) ) = c(Σ) holds if and only if

 1/|γ|   ri (A(x) ) = c(Σ)

for every

(3.12)

γ ∈ C(A(x) ).

(3.13)

i∈¯ γ

Proof. The bounds (3.9) immediately stem from Theorem 2.2 and the Frobenius bounds (1.1). The assertions concerning equalities (3.10) and (3.12) are proved in the same way as those concerning equalities (3.2) and (3.4) in Theorem 3.1, the only difference being that one should use Theorem 1.1 rather than Corollary 5.1 in [14].
Remark 3.2. Note that the upper bound in (3.9) holds true whenever the matrix A¯ is free of zero rows. In the case where the matrix A¯ is reducible (in particular, if it has zero rows), it is reasonable to determine its irreducible components first and then to apply Theorem 3.2 to the corresponding submatrices of the matrices A(x) , x ∈ X. ¯ is empty, then, by [12, Theorem 16.3], the matrix A¯ and, consequently, all the Note also that if the set C(A) matrices A(x) , x ∈ X, are permutationally similar to upper triangular matrices with zero principal diagonal, whence ρ(A(x) ) = 0, x ∈ X. We conclude this subsection by presenting block extensions of Theorems 3.1 and 3.2. These extensions are based on the following two-sided bounds for the Perron root of a block partitioned nonnegative matrix. 1592

Theorem 3.3 [5]. Let A = (aij ) be a nonnegative matrix of order n ≥ 1 and let n =

N 

1 ≤ N ≤ n,

Mi ,

(3.14)

i=1

be a fixed partitioning of the index set into disjoint subsets. Let the nonnegative matrices Aˆ = (ˆ aij )N i,j=1 ,

Aˇ = (ˇ aij )N i,j=1 ,

satisfy the conditions a ˆij ≥ max

k∈Mi




a ˇij ≤ min

akl ,

k∈Mi

l∈Mj




akl ,

i, j = 1, . . . , N.

(3.15)

l∈Mj

Then the Perron root of A satisfies the two-sided bounds ˇ ≤ ρ(A) ≤ ρ(A). ˆ ρ(A)

(3.16)

Now let a finite set Σ = {A(x) }x∈X of nonnegative matrices of order n ≥ 1 be given. Fix a partitioning (3.14) of the index set and define the sets ˆ = {A ˆ(x)}x∈X Σ of N × N nonnegative matrices via the formulas (x) Aˆ(x) = (ˆ aij ),

(x)

a ˆij = max

k∈Mi

and (x) aij ), Aˇ(x) = (ˇ

(x)

a ˇij = min

k∈Mi

ˇ = {Aˇ(x) }x∈X Σ

and




akl ,

(x)

i, j = 1, . . . , N,

x ∈ X,

(3.17)

(x)

i, j = 1, . . . , N,

x ∈ X.

(3.18)

l∈Mj




akl ,

l∈Mj

By Theorem 3.3, we have ρ(Aˇ(x) ) ≤ ρ(A(x) ) ≤ ρ(Aˆ(x) ),

x ∈ X.

(3.19)

ˆ and Σ ˇ and using inequalities Applying the upper and lower bounds of Theorems 3.1 and 3.2 to the matrix sets Σ (3.19), we arrive at the following block generalizations of the two-sided bounds (3.1) and (3.9), where the following notation is used: (x) A¯ = (¯ aij ), where a ¯ij = max a ˆij , i, j = 1, . . . , N, (3.20) x∈X

A = (aij ),

(x)

where aij = max a ˇij , x∈X

i, j = 1, . . . , N.

(3.21)

Theorem 3.4. Let {A(x)}x∈X be a finite set of nonnegative matrices of order n ≥ 1 and let a partitioning ˆ(x) and Aˇ(x) , x ∈ X, be defined in accordance with (3.17) (3.14) of the index set be fixed. Let the matrices A (x) and (3.18), respectively. If the matrices Aˇ , x ∈ X, are free of zero rows, then

α

1−α  (x) (x) min min ri (Aˇ ) min rj (Aˇ ) ≤ ρ(A(x) ) i,j: aij >0

≤

x∈X

(x)

x: ˇ aij >0



max

i,j: a ¯ij >0

α

1−α  (x) (x) max ri (Aˆ ) max rj (Aˆ ) , x∈X

(x)

x: a ˆij >0

(3.22) x ∈ X,

where 0 ≤ α ≤ 1, and min

 s 

min

(x) (i1 ,...,is ,is+1 )∈C(A)  ˇi i >0 j=1 x: a j j+1

≤

max

 s 

max

1/s  rij (Aˇ(x) ) ≤ ρ(A(x) ) 

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) ai i >0 j=1 x: ˆ j j+1

1/s  rij (Aˆ(x) ) , 

(3.23) x ∈ X.

1593

Remark 3.3. In view of Remarks 3.1 and 3.2, the upper bounds in (3.22) and (3.23) hold true under the weaker assumption that the matrix A¯ defined in (3.20) is free of zero rows. Remark 3.4. Note that for the pointwise partitioning n =

n 

{i},

i=1

ˆ(x) = Aˇ(x) , x ∈ X, and the bounds (3.22) and (3.23) of Theorem 3.4 reduce to the we obviously have A(x) = A bounds (3.1) and (3.9) of Theorems 3.1 and 3.2, respectively. On the other hand, in the opposite extreme case where N = 1 and M1 = n, we have Aˆ(x) = max ri(A(x) ),

Aˇ(x) = min ri(A(x) ),

i∈n

i∈n

x ∈ X,

whence the bounds (3.22) and (3.23) reduce to the trivial consequences of the Frobenius bounds min min ri (A(x) ) ≤ ρ(A(x) ) ≤ max max ri(A(x) ),

x∈X i∈n

x∈X i∈n

x ∈ X.

3.2. Joint upper bounds for the spectral radii of complex matrices Let A = (aij ) ∈ Cn×n . Define the nonnegative matrix |A| = (|aij |). The well-known Wielandt lemma (e.g., see [10, p. 358]) asserts that the spectral radius ρ(A) of A satisfies the upper bound ρ(A) ≤ ρ(|A|).

(3.24)

Therefore, given a set Σ = {A(x) }x∈X of complex matrices of order n ≥ 1, we obviously have max ρ(A(x) ) ≤ max ρ(|A(x) |). x∈X

(3.25)

x∈X

In view of (3.25), the value max ρ(A(x) ) can be bounded above by using any joint upper bound for the Perron x∈X

roots of the matrices |A(x) |, x ∈ X, presented in Sec. 3.1. In particular, applying Theorem 3.4 to the set {|A(x)|}x∈X and taking into account Remark 3.3, we arrive at the theorem below. Theorem 3.5. Let {A(x) }x∈X be a finite set of complex matrices of order n ≥ 1 and let a partitioning (3.14) of the index set n be fixed. Denote (x) Aˆ(x) = (ˆ aij ),

(x)

ˆaij = max

k∈Mi




(x)

|akl |,

i, j = 1, . . . , N,

x ∈ X,

(3.26)

l∈Mj

and assume that the matrix A¯ = (¯ aij ),

where

(x)

a ¯ij = max{ˆ aij },

i, j = 1, . . . , N,

(3.27)

α

1−α  (x) ) max rj (Aˆ )

(3.28)

x∈X

is free of zero rows. Then the following joint upper bounds are valid:

(x)

max ρ(A x∈X

)≤

ˆ(x)

max

i,j: ¯ aij >0

max ri (A (x)

and max ρ(A(x) ) ≤ x∈X

1594

x∈X

x: a ˆij >0

max

 s 

¯  (i1 ,...,is ,is+1 )∈C(A) j=1

1/s  max rij (Aˆ(x) ) . (x)  x: a ˆi i >0 j j+1

(3.29)

3.3. The Schur stability of convex combinations of matrices For a given set Σ = {A(x) }x∈X of complex matrices of order n ≥ 1, their convex combinations are matrices of the form
α(x) A(x) , (3.30) x∈X



where α(x) ≥ 0, x ∈ X, and

α(x) = 1.

x∈X

A natural question concerning convex combinations is as follows. Under what conditions on the set Σ all matrices of the form (3.30) are Schur stable? Recall that a matrix is said to be Schur stable if its spectral radius is stricly less than one. This question was considered in [9], where the Schur stability of all convex combinations of k ≥ 2 given matrices was characterized in terms of the so-called block P -matrices. Earlier, the Schur stability problem for convex combinations of two and three matrices was considered in [18] and [8], respectively. Following [13], we consider the more general set of matrices whose rows are independent convex combinations of the corresponding rows of matrices in Σ. More precisely, we consider the set R(Σ) =

C: C=




 T (x) A(x)

,

(3.31)

x∈X (x)

(x)

(x)

where T (x) = diag (t1 , . . . , tn ), ti

≥ 0, i = 1, . . . , n, x ∈ X, and


T (x) = In

(3.32)

x∈X

(In is the identity matrix of order n). For an arbitrary diagonal matrix D with positive diagonal entries, using Wielandt’s lemma, the Frobenius bounds (1.1), and condition (3.32), we derive  ρ




 T

(x)

(x)

A

x∈X

≤ max

i∈n

 =ρ





 T

(x)

D

−1

(x)

A

D

 ≤ max ri i∈n

x∈X




 T

(x)

D

−1

(x)

A

x∈X

D (3.33)

ti ri(D−1 A(x) D) ≤ max max ri (D−1 A(x) D). (x)

i∈n x∈X

x∈X

In view of (3.33), the application of Theorems 2.1 and 2.2 immediately yields the following conditions sufficient for all matrices in R(Σ) to be Schur stable. Theorem 3.6. Let Σ = {A(x)}x∈X be a finite set of complex matrices of order n ≥ 1. Set (x)

¯ = (¯ A aij ),

a ¯ij = max |aij |,

i, j = 1, . . . , n,

x∈X

(3.34)

and assume that the matrix A¯ is free of zero rows. If either for a certain α, 0 ≤ α ≤ 1,

(x)

max

i,j: a ¯ij >0

max ri(A (x)

x: aij =0

or max

 s 

α

1−α  (x) ) max rj (A ) 0

ρ¯(Σ) ≤

max ri (A(x) )

max

max rj (A(x) ) x∈X

(x)

x: aij >0

max

 s 

max

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

1/s  rij (A(x) ) . 

,

(3.50)

(3.51)

Obviously, the condition that the matrix A¯ is free of zero rows is satisfied whenever it is irreducible. If A¯ is reducible, then its rows and columns can be symmetrically permuted to bring A¯ to an upper block triangular matrix having the irreducible components of A¯ as its diagonal blocks. Obviously, the same similarity transformation brings all the matrices A(ω) , ω ∈ X k , k ≥ 1, to the same block triangular form. In view of this, the definition (3.39) implies that ρ(Σ) = max ρ(Σ[Mi ]), 1≤i≤N

where Mi ⊆ n, i = 1, . . . , N , are the disjoint subsets of the index set n corresponding to the irreducible  ¯ and we denote Σ[Mi ] = A(x) [Mi ] components of the matrix A, . Thus, the case of a general matrix A¯ x∈X reduces to the irreducible case. In order to derive lower bounds for the lower spectral radius of a set of nonnegative matrices, first, we establish the following simple lower bound.   Lemma 3.1. Let Σ = A(x) x∈X be a finite set of nonnegative matrices. Then ρ(Σ) ≥ min min ri (A(x) ).

(3.52)

A(ω) ∞ = A(x1 ) · · · A(xk ) ∞ ≥ min ri(A(x1 ) · · · A(xk ) ).

(3.53)

x∈X i∈n

Proof. For ω = x1 . . . xk , we derive i∈n

1597

For nonnegative n × n matrices B and C, we obviously have ri (BC) = (BCe)i ≥ min {rj (C)} (Be)i = ri(B) min {rj (C)}, j∈n

i = 1, . . . , n,

j∈n

where e = [1, . . . , 1]T . Therefore, min ri(A(x1 ) · · · A(xk ) ) ≥

i∈n

k  j=1



k min ri(A(xj ) ) ≥ min min ri(A(x) ) . x∈X i∈n

i∈n

(3.54)

Inequalities (3.53) and (3.54) imply that (x) A(ω) 1/k ), ∞ ≥ min min ri (A

ω ∈ Xk ,

x∈X i∈n

k ≥ 1.

(3.55)

Finally, inequality (3.55) and the definition (3.42) of the lower spectral radius, which is independent of the norm used, prove (3.52).
In view of equality (3.45), from Lemma 3.1 it follows that for an arbitrary nonsingular diagonal matrix D, the following lower counterpart of the upper bound (3.49) is valid: ρ(Σ) ≥ min min ri(D−1 A(x) D).

(3.56)

x∈X i∈n

Now, as in the case of the upper bound for the joint spectral radius, lower bounds for the lower spectral radius of a set of nonnegative matrices are obtained by applying the lower bounds of Theorems 2.1 and 2.2. In this way, we arrive at the next result. Theorem 3.8. Let Σ = {A(x)}x∈X be a finite set of nonnegative matrices of order n ≥ 1. If all the matrices A(x) , x ∈ X, are free of zero rows, then the lower spectral radius ρ(Σ) satisfies the lower bounds

ρ(Σ) ≥

α

1−α  min ri(A(x) ) min rj (A(x) )

min

i,j: a ¯ij >0

and ρ(Σ) ≥

x∈X

(x)

x: aij >0

min

 s 

min

(x) ¯  (i1 ,...,is ,is+1 )∈C(A) j=1 x: aij ij+1 >0

1/s  rij (A(x) ) . 

(3.57)

(3.58)

Here, the matrix A¯ = (¯ aij ) is defined as in Theorem 3.7. We conclude this section by considering, for completeness, an alternative approach to deriving upper bounds for the joint spectral radius of a set of complex matrices and lower bounds for the lower spectral radius of a set of nonnegative matrices. Given a finite set Σ = {A(x) }x∈X of complex matrices of order n ≥ 1, define the nonnegative matrices ¯ A = (¯ aij ) and A = (aij ) by the relations (x)

¯aij = max |aij |, x∈X

(x)

aij = min |aij |, x∈X

i, j = 1, . . . , n.

(3.59)

Then we obviously have the componentwise matrix inequalities ¯ A ≤ |A(x)| ≤ A,

x ∈ X,

whence Ak ≤ |A(x1 ) | · · · |A(xk) | ≤ A¯k ,

x1 , . . . , xk ∈ X,

k ≥ 1.

By Wielandt’s lemma (e.g., see [10, p. 358]), the right-hand inequality in (3.60) implies that ¯ ρ(A(ω) )1/k ≤ ρ(A), 1598

ω ∈ Xk ,

k ≥ 1,

(3.60)

and, in view of the definition (3.39), we have ¯ ρ(Σ) ≤ ρ(A).

(3.61)

Note that inequality (3.61), first suggested in [4] for nonnegative matrices, improves the earlier bound [3]  ρ(Σ) ≤ ρ




 (x)

A

,

x∈X

which is valid for finite sets of nonnegative matrices. Quite similarly, for a finite set Σ of nonnegative matrices, based on equality (3.44), from the left-hand inequality in (3.60) we derive ρ(Σ) ≥ ρ(A). (3.62) In view of (3.61) and (3.62), a natural way to bound ρ(Σ) ¯ and ρ(Σ) from above and below, respectively, is to use known upper and lower bounds for the Perron root of a single nonnegative matrix. In this way, by applying Corollary 3.1 and Theorem 1.1, we obtain the following bounds:

ρ¯(Σ) ≤

max



i,j: a ¯ij >0

 ¯ α rj (A) ¯ 1−α , ri(A)

ρ¯(Σ) ≤ max

¯  γ∈C(A) i∈¯ γ

and ρ(Σ) ≥

min

i,j: aij >0

  ri (A)α rj (A)1−α ,

 

ρ(Σ) ≥ min

 

γ∈C(A)  i∈¯ γ

1/|γ|  ¯ ri (A) ; 

1/|γ|  ri (A) . 

(3.63)

(3.64)

The bounds (3.63) hold for a finite set of complex matrices such that the matrix A¯ is free of zero rows, and the bounds (3.64) hold for a finite set of nonnegative matrices such that the matrix A is free of zero rows. Comparing the bounds (3.63) with the bounds of Theorem 3.7, we readily see that the former are implied by the latter. Note also that the bounds (3.57) and (3.58) are applicable under somewhat weaker conditions than the bounds (3.64). Moreover, as the following example demonstrates, in some cases, the bounds of Theorems 3.7 and 3.8 are better than the bounds (3.61) and (3.62). Indeed, for n ≥ 2, let  A = (aij ),

aij =

1 if j = i + 1, 1 ≤ i ≤ n − 1 or i = n, j = 1; 0 otherwise,

and consider the set Σ = {A, A2 , . . . , An }. As is not difficult to ascertain, all the bounds (3.50), (3.51), (3.57), and (3.58) are applicable and yield 1 ≤ ρ(Σ) ≤ ρ¯(Σ) ≤ 1, implying that ρ(Σ) = ρ¯(Σ) = 1, whence all the bounds of Theorems 3.7 and 3.8 hold with equality. On the other hand, it can readily be seen that A = 0,

A¯ = e eT ,

whence ρ(A) = 0,

¯ = n. ρ(A)

Thus, for the example considered, the lower bound (3.62) is always trivial, whereas the upper bound (3.61) becomes less and less accurate as n increases. Translated by L. Yu. Kolotilina. 1599

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