Reducible Matrices

Nonnegative Matrices and Stochastic Matrices 9-7

With D=I , a relaxation of this bound onτ(P ) yields the expression

≤min

⎧⎨

⎩ρ− n

j=1

mini

P_{i j}v_j v_i

,

n j=1

maxi

P_{i j}v_j v_i

−ρ

⎫⎬

⎭.

5. [RT85, Theorem 4.3] For a positive vector u ∈Rⁿ, consider the function M^u:Rⁿ →Rdefined for a∈Rⁿby

M^u(a)=max{x^Ta : x∈Rⁿ,x ≤1, x^Tu=0}.

This function has a simple explicit representation obtained by sorting the ratios^a_u^j

j, i.e., identifying a permutation j (1),. . ., j (n) of 1,. . ., n such that

aj (1)

uj (1)

≤aj (2)

uj (2)

≤ · · · ≤ aj (n)

uj (n)

.

With kas the smallest integer in{1,. . ., n}such that 2^k_p=1 uj ( p)>ⁿ_t=1utand

µ≡1+

⎛

⎝ⁿ

t=1

ut−2

k

p=1

uj ( p)

⎞

⎠,

we have that

M^u(a)=

k−1 p=1

aj ( p)+µaj (k)− n p=k+1

aj ( p).

With.as the∞-norm onRⁿand (D⁻¹P D)₁,. . ., (D⁻¹P D)_nas the columns of D⁻¹P D, the bound in Fact 11 on the coefficient of ergodicityτ(P ) of P becomes

rmax=1,...,nM^D−1w[(D⁻¹P D)_r].

P is convergent or transient if lim_m→∞P^m=0.

P is semiconvergent if limm→∞P^mexists.

P is weakly expanding if P u≥u for some u>0.

P is expanding if for some P u>u for some u>0.

An n×n matrix polynomial of degree d in the (integer) variable m is a polynomial in m with coefficients that are n×n matrices (expressible as S(m)=^d_t=0m^tBtwith B1,. . ., Bdas n×n matrices and Bd=0).

Facts:

Facts requiring proofs for which no specific reference is given can be found in [BP94, Chap. 2].

1. The set of basic classes of a nonnegative matrix is always nonempty.

2. (Spectral Properties of the Perron Value) Let P be a nonnegative n×n matrix with spectral radius ρand indexν.

(a) [Fro12]ρis an eigenvalue of P .

(b) [Fro12] There exist semipositive right and left eigenvectors of P corresponding toρ, i.e.,ρis a distinguished eigenvalue of both P and P^T.

(c) [Rot75]νis the largest number of vertices on a simple walk in R^∗(P ).

(d) [Rot75] For each basic class B having height h, there exists a generalized eigenvector v^B in N_ρ^h(P ), with (v^B)i>0 if i →B and (v^B)i =0 otherwise.

(e) [Rot75] The dimension of N_ρ^ν(P ) is the number of basic classes of P . Further, if B1,. . ., Bp

are the basic classes of P and v^B1,. . ., v^Brare generalized eigenvectors of P atρthat satisfy the conclusions of Fact 2(d) with respect to B1,. . ., Br, respectively, then v^B1,. . ., v^Bpform a basis of N_ρ^ν(P ).

(f) [RiSc78, Sch86] If B1,. . ., Bpis an enumeration of the basic classes of P with nondecreasing heights (in particular, s <t assures that we do not have Bt→Bs), then there exist generalized eigenvectors v^B1,. . ., v^Bp of P atρthat satisfy the assumptions and conclusions of Fact 2(e) and a nonnegative p×p upper triangular matrix M with all diagonal elements equal toρ, such that

P [v^B1,. . ., v^Bp]=[v^B1,. . ., v^Bp]M

(in particular, v^B1,. . ., v^Bp is a basis of N_ρ^ν(P )). Relationships between the matrix M and the Jordan Canonical Form of P are beyond the scope of the current review; see [Sch56], [Sch86], [HS89], [HS91a], [HS91b], [HRS89], and [NS94].

(g) [Vic85], [Sch86], [Tam04] If B1,. . ., Brare the basic classes of P having height 1 and v^B1,. . ., v^Br are generalized eigenvectors of P atρthat satisfy the conclusions of Fact 2(d) with respect to B₁,. . ., B_r, respectively, then v^B1,. . ., v^Brare linearly independent, nonnegative eigenvectors of P atρthat span the cone (_R⁺₀)ⁿ∩N_ρ¹(P ); that is, each vector in the cone (_R⁺₀)ⁿ∩N_ρ¹(P ) is a linear combination with nonnegative coefficients of v^B1,. . ., v^Br(in fact, the sets{αv^Bs:α≥0} for s =1,. . ., r are the the extreme rays of the cone (R⁺₀)ⁿ∩N_ρ¹(P )).

3. (Spectral Properties of Eigenvaluesλ=ρ(P ) with|λ| =ρ(P )) Let P be a nonnegative n×n matrix with spectral radiusρ, indexν, co-index ¯ν, period q , and coefficient of ergodicityτ.

(a) [Rot81a] The following are equivalent:

i. {λ∈σ(P )\ {ρ}:|λ| =ρ} = ∅.

ii. ¯ν=0.

iii. P is aperiodic (q=1).

Nonnegative Matrices and Stochastic Matrices 9-9 (b) [Rot81a] Ifλ ∈ σ(P )\ {ρ}and|λ| = ρ, then (^λ_ρ)^h = 1 for some h ∈ {2,. . ., n}; further, q = min{h = 2,. . ., n : (_ρ^λ)^h = 1 for eachλ ∈ σ(P )\ {ρ}with|λ| = ρ} ≤ n (here the minimum over the empty set is taken to be 1).

(c) [Rot80] Ifλ∈σ(P )\{ρ}and|λ| =ρ, thenνP(λ) is bounded by the largest number of vertices on a simple walk in R^∗(P ) with each vertex corresponding to a (basic) access equivalence class C that hasλ∈σ(P [C ]); in particular, ¯ν≤ν.

4. (Distinguished Eigenvalues) Let P be a nonnegative n×n matrix.

(a) [Vic85]λis a distinguished eigenvalue of P if and only if there is a final set C withρ(P [C ])=λ.

It is noted that the set of distinguished eigenvalues of P and P^Tneed not coincide (and the above characterization of distinguished eigenvalues is not invariant of the application of the transpose operator). (See Example 1 below.)

(b) [HS88b] Ifλis a distinguished eigenvalue,νP(λ) is the largest number of vertices on a simple walk in R^∗(P [λ]).

(c) [HS88b] Ifµ >0, thenµ≤min{λ:λis a distinguished eigenvalue of P}if and only if there exists a vector u>0 with P u≥µu.

(For additional characterizations of the minimal distinguished eigenvalue, see the concluding remarks of Facts 12(h) and 12(i).)

Additional properties of distinguished eigenvaluesλof P that depend on P [λ] can be found in [HS88b] and [Tam04].

5. (Convergence Properties of Powers) Let P be a nonnegative n×n matrix with positive spectral radius ρ, indexν, co-index ¯ν, period q , and coefficient of ergodicityτ(for the case whereρ=0, see Fact 12(j) below).

(a) [Rot81a] There exists an n×n matrix polynomial S(m) of degreeν−1 in the (integer) variable m such that limm→∞[(^P_ρ)^m−S(m)]=0 (C, p) for every p≥ν¯; further, if P is aperiodic, this limit holds as a regular limit and the convergence is geometric with rate_ρ^τ <1.

(b) [Rot81a] There exist matrix polynomials S⁰(m),. . ., S^q−1(m) of degreeν−1 in the (integer) variable m, such that for each k = 0,. . ., q−1, limm→∞[(^P_ρ)^mq+k −S^t(m)] = 0 and the convergence of these sequences to their limit is geometric with rate (^τ_ρ)^q <1.

(c) [Rot81a] There exists a matrix polynomial T (m) of degreeνin the (integer) variable m with lim_m→∞[^m_s=⁻₀¹(^P_ρ)^s −T (m)]=0 (C, p) for every p≥ν; further, if P is aperiodic, this limit¯ holds as a regular limit and the convergence is geometric with rate^τ_ρ <1.

(d) [FrSc80] The limit of_ρm^P_m^m_ν−1[I+ ^P_ρ + · · · +(^P_ρ)^q−1] exists and is semipositive.

(e) [Rot81b] Let x =[xi] be a nonnegative vector inRⁿand let i ∈ n. With K (i, x) ≡ {j ∈ n: j→i} ∩ {j ∈ n: u→j for some u∈ nwith xu>0},

r (i|x, P )≡inf{α >0 : lim

m→∞α^−m(P^mx)i=0} =ρ(P [K (i, x)]) and if r ≡r (i|x, P )>0,

k(i|x, P )≡inf{k =0, 1,. . . : lim

m→∞m^−kr^−m(P^mx)i =0} =νP [K (i,x)](r ).

Explicit expressions for the polynomials mentioned in Facts 5(a) to 5(d) in terms of characteristics of the underlying matrix P are available in Fact 12(a)ii for the case whereν=1 and in [Rot81a]

for the general case. In fact, [Rot81a] provides (explicit) polynomial approximations of additional high-order partial sums of normalized powers of nonnegative matrices.

6. (Bounds on the Perron Value) Let P be a nonnegative n×n matrix with spectral radiusρand letµ be a nonnegative scalar.

(a) For ∈ {<,≤,=,≥,>},

[P uµu for some vector u>0]⇒[ρµ] ; further, the inverse implication holds foras<, implying that

ρ=max

x⁰ min

{i : xi>0}

( Ax)i

xi . (b) For ∈ {,≤,=,≥,},

[ρµ]⇒[P uµu for some vector u0] ;

further, the inverse implication holds foras≥. (c) ρ < µif and only if P u< ρu for some vector u≥0 .

Sinceρ(P^T) = ρ(P ), the above properties (and characterizations) of ρ can be expressed by applying the above conditions to P^T. (See Example 3 below.)

Some of the above results can be expressed in terms of the Collatz–Wielandt sets. (See Fact 7 of Section 9.2 and Chapter 26.)

7. (Bounds on the Spectral Radius) Let P be a nonnegative n×n matrix and let A be a complex n×n matrix such that|A| ≤P . Thenρ( A)≤ρ(P ).

8. (Functional Inequalities) Consider the functionρ(.) mapping nonnegative n×n matrices to their spectral radius.

(a) ρ(.) is nondecreasing in each element (of the domain matrices); that is, if A and B are non-negative n×n matrices with A≥B≥0, thenρ( A)≥ρ(B ).

(b) [Coh78]ρ(.) is (jointly) convex in the diagonal elements; that is, if A and D are n×n matrices, with D diagonal, A and A+D nonnegative, and if 0< α <1, thenρ[αA+(1−α)( A+D)]≤

αρ( A)+(1−α)ρ( A+D).

(c) [EJD88] If A=[ai j] and B=[bi j] are nonnegative n×n matrices, 0< α <1 and C =[ci j] with ci j =a_{i j}^αb¹_{i j}^−αfor each i, j=1,. . ., n, thenρ(C )≤ρ( A)^αρ(B )^1−α.

Further functional inequalities aboutρ(.) can be found in [EJD88] and [EHP90].

9. (Resolvent Expansions) Let P be a nonnegative square matrix with spectral radiusρand letµ > ρ.

ThenµI−P is invertible and

(µI−P )⁻¹=^∞

t=0

P^t

µ^t+1 ≥ I

µ+ P

µ² ≥ I

µ ≥0

(the invertibility ofµI−P and the power series expansion of its inverse do not require nonnegativity of P ).

For explicit expansions of the resolvent about the spectral radius, that is, for explicit power series representations of [(z+ρ)I −P ]⁻¹with|z|positive and sufficiently small, see [Rot81c], and [HNR90] (the latter uses such expansions to prove Perron–Frobenius-type spectral results for nonnegative matrices).

10. (Puiseux Expansions of the Perron Value) [ERS95] The functionρ(.) mapping irreducible non-negative n×n matrices X = [xi j] to their spectral radius has a converging Puiseux (fractional power series) expansion at each point; i.e., if P is a nonnegative n×n matrix and if F is an n×n matrix with P+F ≥0 for all sufficiently small positive, thenρ(P +F ) has a representation _∞

k=0ρk^k/qwithρ0=ρ(P ) and q as a positive integer.

11. (Bounds on the Ergodicity Coefficient) [RT85, extension of Theorem 3.1] Let P be a nonnegative n×n matrix with spectral radiusρ, corresponding semipositive right eigenvector v, and ergodicity

Nonnegative Matrices and Stochastic Matrices 9-11 coefficientτ, let D be a diagonal n×n matrix with positive diagonal elements, and let.be a norm onRⁿ. Then

τ ≤ max

x∈R^n,x≤1,x^TD⁻¹v=0

x^TD⁻¹P D.

12. (Special Cases) Let P be a nonnegative n×n matrix with spectral radiusρ, indexν, and period q . (a) (Index 1) Supposeν=1.

i. ρI−P has a group inverse.

ii. [Rot81a] With P≡I−(ρI−P )(ρI−P )^#, all of the convergence properties stated in Fact 6 of Section 9.2 apply.

iii. Ifρ >0, then ^P_ρm^mis bounded in m (element-wise).

iv. ρ=0 if and only if P =0.

(b) (Positive eigenvector) The following are equivalent:

i. P has a positive right eigenvector corresponding toρ. ii. The final classes of P are precisely its basic classes.

iii. There is no vector w satisfying w^TP ρw^T. Further, when the above conditions hold:

i. ν=1 and the conclusions of Fact 12(a) hold.

ii. If P satisfies the above conditions and P = 0, thenρ > 0 and there exists a diagonal matrix D having positive diagonal elements such that S≡ _ρ¹D⁻¹P D is stochastic (that is, S≥0 and S1=1; see Chapter 4).

(c) [Sch53] There exists a vector x>0 with P x≤ρx if and only if every basic class of P is final.

(d) (Positive generalized eigenvector) [Rot75], [Sch86], [HS88a] The following are equivalent:

i. P has a positive right generalized eigenvector atρ.

ii. Each final class of P is basic.

iii. P u≥ρu for some u>0.

iv. Every vector w≥0 with w^TP ≤ρw^T must satisfy w^TP=ρw^T. v. ρis the only distinguished eigenvalue of P .

(e) (Convergent/Transient) The following are equivalent:

i. P is convergent.

ii. ρ <1.

iii. I−P is invertible and (I−P )⁻¹≥0.

iv. There exists a positive vector u∈Rⁿwith P u<u.

Further, when the above conditions hold, (I−P )⁻¹=^∞t=0P^t≥I . (f) (Semiconvergent) The following are equivalent:

i. P is semiconvergent.

ii. Eitherρ <1 orρ=ν=1 and 1 is the only eigenvalueλof P with|λ| =1.

(g) (Bounded) P^mis bounded in m (element-wise) if and only if eitherρ <1 orρ=1 andν=1.

(h) (Weakly Expanding) [HS88a], [TW89] [DR05] The following are equivalent:

i. P is weakly expanding.

ii. There is no vector w∈Rⁿwith w≥0 and w^TP w^T. iii. Every distinguished eigenvalueλof P satisfiesλ≥1.

iv. Every final class C of P hasρ(P [C ])≥1.

v. If C is a final set of P , thenρ(P [C ])≥1.

Givenµ >0, the application of the above equivalence to^P_µyields characterizations of instances where each distinguished eigenvalue of P is bigger than or equal toµ.

(i) (Expanding) [HS88a], [TW89] [DR05] The following are equivalent:

i. P is expanding.

ii. There exists a vector u∈Rⁿwith u≥0 and P u>u.

iii. There is no vector w∈Rⁿwith w0 and w^TP ≤w^T. iv. Every distinguished eigenvalueλof P satisfiesλ >1.

v. Every final class C of P hasρ(P [C ])>1.

vi. If C is a final set of P , thenρ(P [C ])>1.

Givenµ >0, the application of the above equivalence to^P_µyields characterizations of instances where each distinguished eigenvalue of P is bigger thanµ.

(j) (Nilpotent) The following are equivalent conditions:

i. P is nilpotent; that is, P^m=0 for some positive integer m.

ii. P is permutation similar to an upper triangular matrix all of whose diagonal elements are 0.

iii. ρ=0.

iv. Pⁿ=0.

v. P^ν=0.

(k) (Symmetric) Suppose P is symmetric.

i. ρ=maxu^0uu^{TTP u}u.

ii. ρ=^u_u^TT^{P u}u for u0 if and only if u is an eigenvector of P corresponding toρ. iii. [CHR97, Theorem 1] For u, w0 with wi =√

ui(P u)ifor i =1,. . ., n,^u_u^TT^{P u}u ≤ ^w_w^TT^{P w}w

and equality holds if and only if u[S] is an eigenvector of P [S] corresponding toρ, where S≡ {i : ui>0}.

Examples:

1. We illustrate parts of Fact 2 using the matrix

P =

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

2 2 2 0 0 0

0 2 0 0 0 0

0 0 1 2 0 0

0 0 0 1 1 0

0 0 0 1 1 1

0 0 0 0 0 1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦ .

The eigenvalues of P are 2,1, and 0; so, ρ(P ) = 2 ∈ σ(P ) as is implied by Fact 2(a). The vectors v =[1, 0, 0, 0, 0, 0]^T and w=[0, 0, 0, 1, 1, 1] are semipositive right and left eigenvectors corresponding to the eigenvalue 2; their existence is implied by Fact 2(b).

The basic classes are B1 = {1}, B1 = {2}and B3 = {4, 5}. The digraph corresponding to P , its reduced digraph, and the basic reduced digraph of P are illustrated in Figure 9.1. From Figure 9.1(c), the largest number of vertices in a simple walk in the basic reduced digraph of P is 2 (going from B1

to either B2or B3); hence, Fact 2(c) implies thatνP(2)=2. The height of basic class B1is 1 and the height of basic classes B2and B3is 2. Semipositive generalized eigenvectors of P at (the eigenvalue)

Nonnegative Matrices and Stochastic Matrices 9-13

5 3 4

(a) (b) (c)

1

2 {3}

{4,5}

{1}

{2}

{6}

{4,5}

{1}

{2}

6

FIGURE 9.1 (a) The digraph(P ), (b) reduced digraph R[(P )], and (c) basic reduced digraph R^∗(P ).

2 that satisfy the assumptions of Fact 2(f) are u^B1 =[1, 0, 0, 0, 0, 0]^T, u^B2 =[1, 1, 0, 0, 0, 0]^T, and u^B3 =[1, 0, 2, 1, 1, 0]^T. The implied equality

P [u^B1,. . ., u^Bp]=[u^B1,. . ., u^Bp]M of Fact 2(f) holds as

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

2 2 2 0 0 0

0 2 0 0 0 0

0 0 1 2 0 0

0 0 0 1 1 0

0 0 0 1 1 1

0 0 0 0 0 1

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

1 1 1

0 1 0

0 0 2

0 0 1

0 0 1

0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

2 4 6

0 2 0

0 0 4

0 0 2

0 0 2

0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

=

⎡

⎢⎢

⎢⎢

⎢⎢

⎢⎣

1 1 1

0 1 0

0 0 2

0 0 1

0 0 1

0 0 0

⎤

⎥⎥

⎥⎥

⎥⎥

⎥⎦

⎡

⎢⎣

2 2 4

0 2 0

0 0 2

⎤

⎥⎦.

In particular, Fact 2(e) implies that u^B1, u^B2, u^B3 form a basis of N_ρ^ν_{(P )}^{(P )} = N₂². We note that while there is only a single basic class of height 1, dim[N_ρ¹(P )] = 2 and u^B1, 2u^B2 −u^B3 = [−1, 2,−2,−1,−1, 0]^T form a basis of N_ρ¹(P ). Still, Fact 2(g) assures that (R⁺₀)ⁿ∩N_ρ¹(P ) is the cone{αu^B1 :α≥0}(consisting of its single ray).

Fact 4(a) and Figure 9.1 imply that the distinguished eigenvalues of P are 1 and 2, while 2 is the only distinguished eigenvalue of P^T.

2. Let H = 0 1

1 0

; properties of H were demonstrated in Example 2 of section 9.2. We will demon-strate Facts 2(c), 5(b), and 5(a) on the matrix

P ≡

H I

0 H

.

The spectral radius of P is 1 and its basic classes of P are B1 = {1, 2}and B2 = {3, 4}with B1

having access to B2. Thus, the index of 1 with respect to P , as the largest number of vertices on a walk of the marked reduced graph of P , is 2 (Fact 2(c)). Also, as the period of each of the two basic

classes of P is 2, the period of P is 2. To verify the convergence properties of P , note that

P^m=

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

I mH

0 I

if m is even

H mI

0 H

if m is odd,

immediately providing matrix–polynomials S⁰(m) and S¹(m) of degree 1 such that limm→∞P^2m− S⁰(m)=0 and limm→∞P^2m+1−S¹(m)=0. In this example,τ(P ) is 0 (as the maximum over the empty set) and the convergence of the above sequences is geometric with rate 0.

The above representation of P^mshows that P^m=

H^m mH^m+1

0 H^m

and Example 2 of section 9.2 shows that

mlim→∞H^m= I+H

2 =

.5 .5

.5 .5

(C,1).

We next consider the upper-right blocks of P^m. We observe that 1

m

m−1

t=0

P^t[B1, B2]= _mI

4 +^(m−₄^2)H if m is even

(m−1)²I

4m +^(m2_4m^−1)H if m is odd,

=

_m(I+H)

4 − ^H₂ if m is even

m(I+H)

4 −₂^I + ^I−_4m^H if m is odd, implying that

mlim→∞

1 m

m−1

t=0

P^t[B1, B2]−m

I+H 4

+ I+H

4 =0 (C,1).

As m−1= _m^1m−1_t=0 t for each m=1, 2,. . ., the above shows that

m→∞lim 1 m

m−1

t=0

P^t[B1, B2]−t

I+H

4 =0 (C,1),

and, therefore (recalling that (C,1)-convergence implies (C,2)-convergence),

mlim→∞

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ P^m−

⎡

⎢⎢

⎢⎣

.5 .5 −.25m −.25m

.5 .5 −.25m −.25m

0 0 .5 .5

0 0 .5 .5

⎤

⎥⎥

⎥⎦

⎫⎪

⎪⎪

⎬

⎪⎪

⎪⎭

=0 (C,2).

3. Fact 6 implies many equivalencies, in particular, as the spectral radius of a matrix equals that of its transpose. For example, for a nonnegative n×n matrix P with spectral radiusρand nonnegative scalarµ, the following are equivalent:

(a) ρ < µ.

(b) P u< µu for some vector u>0.

(c) w^TP < µw^Tfor some vector w>0.

Nonnegative Matrices and Stochastic Matrices 9-15 (d) P u< ρu for some vector u≥0.

(e) w^TP < ρw^Tfor some vector w≥0.

(f) There is no vector u0 satisfying P u≥µu.

(g) There is no vector w0 satisfying w^TP ≥µw^T.

No documento The Editor (páginas 148-156)