CONTENTS 1 A MOTIVATING EXAMPLE: FLOCKING
Lecture: linear algebra
Sonia Mart´ınez
April 12, 2018
Abstract The treatment corresponds to selected parts from Chapter 1 in [1] and from Chapter 2 in [2].
Contents
1 A motivating example: flocking 1
2 Matrix concepts 2 2.1 Basic notation ...... 2 2.2 Discrete-time linear systems and (semi-)convergent matrices ...... 3 2.3 Special matrix sets ...... 3 2.4 Eigenvalues, singular values, and induced norms ...... 5 2.5 Spectral radius and general criterion for convergent matrices ...... 5
3 Birkho↵–von Neumann theorem 6
4 Perron-Frobenius theory 7 4.1 Positive matrices ...... 8 4.2 Nonnegative matrices ...... 9 4.3 Application to dynamical systems ...... 11
1 A motivating example: flocking
Consider n agents all moving in the plane with the same speed but with di↵erent headings. Each agent’s heading is updated using a simple local rule based on the average of its own heading plus the headings of its neighbors. For our purposes here, let us assume that the interaction topology is fixed, i.e., the set of neighbors of each agent remains the same throughout the evolution.
The heading of agent i 1,...,n ,written✓i, evolves in discrete time in accordance with a model of the form 2{ } 1 ✓i(t + 1) = ✓i(t)+ ✓j(t) . (1) 1+di 0 1 j X2Ni @ A 1
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2 MATRIX CONCEPTS
Here, i denotes the set of neighbors of agent i, and di is the cardinality of this set. Headings are represented as realN numbers in the interval [0, 2⇡). If we think of this model as an algorithm, then certainly this algorithm is distributed, in the sense that each agent only needs to interact with its neighbors in the graph to execute it.
Exercise 1.1 Ponder a bit about the advantages and disadvantages of interpreting a heading as a number in [0, 2⇡). • As long as the interaction graph is connected, it turns out that the algorithm (1) will make all agents agree on the same heading and “flock” in one direction. Even more, even if the interaction topology is changing between timesteps (say, at time t, agents i and j are neighbors, but they are not at time t + 1), one can show that, as long as it remains connected, the same flocking behavior emerges. Even more, for dynamic interaction topologies, they might not be connected ever, but as long as they are not “too disconnected” (and we will make this very loose statement fully precise in the future), flocking will be achieved. This example will serve us as a guide in the introduction of several important concepts in this lecture. It is instructive to write the heading update equations (1) in compact form, ✓(t + 1) = F✓(t), (2) where F is the matrix with elements 1 Fii = , 1+di 1 1+d j is a neighbor of i, Fij = i (0 j is not a neighbor of i. Several properties of F are worth noticing. First, the matrix is square and nonnegative (i.e., all its elements are greater than or equal to zero). Second, the rows of F all sum up to one. These properties play a critical role in explaining the convergence properties of the algorithm (1).
2 Matrix concepts
2.1 Basic notation
We let C, R, N, and Z denote the set of complex, real, natural and integer numbers, respectively; also R 0 and Z 0 are the set of non-negative real numbers and non-negative integer numbers. For a
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2.2 Discrete-time linear systems and (semi-)convergent matrices 2 MATRIX CONCEPTS
2.2 Discrete-time linear systems and (semi-)convergent matrices
Definition 2.1 (Discrete-time linear system) A square matrix A defines a discrete-time linear system by x(k + 1) = Ax(k),x(0) = x0, (3) k or, equivalently by x(k)=A x0, where the sequence x(k) k Z 0 is called the solution, trajectory or evolution of the system. { } 2
Sometimes it is convenient to adopt the shorthand x+ = f(x) to denote the system x(k + 1) = f(x(k)), for a given map f : Rn Rn. We are interested in understanding when a solution from an arbitrary initial condition has an asymptotic! limit as time diverges and to what value the solution converges. We formally de ne this property as follows.
n n Definition 2.2 (Convergent and semi-convergent matrices) AmatrixA R ⇥ is 2 ` (i) semi-convergent if lim` + A exists; and ! 1 ` (ii) convergent if it is semi-convergent and lim` + A =0. ! 1 • k It is immediate to see that, if A is semi-convergent with limiting matrix A =limk + A ,thenlimk + x(k)= 1 ! 1 ! 1 A x0. 1 Remark 2.3 (Modal decomposition for symmetric matrices) Before treating the general analysis method, we present a self-contained and instructive case of symmetric matrices. Recall that a symmetric matrix A has real eigenvalues 1 2 n and corresponding orthonormal (i.e., orthogonal and unit-length) ··· n eigenvectors v1,...,vn. Because the eigenvectors are an orthonormal basis for R , we can write the modal th decomposition x(k)=y (k)v + + y (k)v ,wherethei normal mode is defined by y (k)=v>x(k). 1 1 ··· n n i i We then left-multiply the two equalities (3)byvi> and exploit Avi = ivi to obtain yi(k + 1) = iyi(k), k yi(0) = vi>x0,whichimpliesyi(k)= i (vi>x0). In short, the evolution of the linear system is
k k x(k)= (v>x )v + + (v>x )v . 1 1 0 1 ··· n n 0 n Therefore, each evolution starting from an arbitrary initial condition satisfies
(i) limk x(k)=0n if and only if i < 1 for all i 1,...,n and !1 | | 2{ }
(ii) limk x(k)=(v1>x0)v1 + +(vm> x0)vm if and only if 1 = = m = 1 and i < 1 for all i !1m +1,...,n ··· ··· | | 2{ }
2.3 Special matrix sets
The provided examples of MAS are defined via special matrices, whose properties are key to establish whether they are convergent or not. In the following we recall a few definitions. n n A matrix A R ⇥ with entries aij, i, j 1,...,n ,is 2 2{ } T (i) Orthogonal if AA = In, and is special orthogonal if it is orthogonal with det(A) = +1. The set of orthogonal matrices is a group.1
1AsetG with a binary operation, denoted by G G (a, b) a?b G,isagroup if: (i) a?(b?c)=(a?b) ?c for all a, b, c G (associativity property); (ii) there exists e⇥ G3such that7! a?e2= e?a= a for all a G (existence of an identity element);2 and (iii) there exists a 1 G such that a?a2 1 = a 1 ?a= e for all a G (existence of2 inverse elements). 2 2 3
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2.3 Special matrix sets 2 MATRIX CONCEPTS
(ii) Nonnegative (resp., positive) if all its entries are nonnegative (resp., positive).
n (iii) Row-stochastic (or stochastic for brevity) if it is nonnegative and j=1 aij = 1, for all i 1,...,n ; in other words, A is row-stochastic if 2{ } P A1n = 1n.
(iv) Column-stochastic if it is nonnegative and n a = 1, for all j 1,...,n . i=1 ij 2{ } (v) Doubly stochastic if A is row-stochastic andP column-stochastic. (vi) Normal if AT A = AAT . (vii) A permutation matrix if A has precisely one entry equal to one in each row, one entry equal to one in each column, and all other entries equal to zero. The set of permutation matrices is a group.
Example 2.4 (Toeplitz and Circulant matrices) Toeplitz matrices are square matrices with equal en- tries along each diagonal parallel to the main diagonal. In other words, a Toeplitz matrix is a matrix of the form ...... t0 t1 . . . tn 2 tn 1 2 ...... 3 t 1 t0 t1 . . . tn 2 6 ...... 7 6 . t 1 t0 t1 . . . 7 6 7 6 ...... 7 . 6 . . t 1 t0 t1 . . 7 6 7 6 ...... 7 6 . . . t 1 t0 t1 . 7 6 7 6 . . . 7 6t ...... t t t 7 6 n+2 1 0 1 7 6 . . . 7 6 ...... 7 6t n+1 t n+2 t 1 t0 7 4 5 An n n Toeplitz matrix is determined by its first row and column, and hence by 2n 1 scalars. ⇥ Circulant matrices are square Toeplitz matrices where each two subsequent row vectors vi and vi+1 have the following two properties: the last entry of vi is the first entry of vi+1 and the first (n 1) entries of vi are the second (n 1) entries of v . In other words, a circulant matrix is a matrix of the form i+1 ...... c0 c1 . . . cn 2 cn 1 2 ...... 3 cn 1 c0 c1 . . . cn 2 6 ...... 7 6 . cn 1 c0 c1 . . . 7 6 7 6 ...... 7 , 6 . . cn 1 c0 c1 . . 7 6 7 6 ...... 7 6 . . . cn 1 c0 c1 . 7 6 7 6 . . . 7 6 c ...... c c c 7 6 2 n 1 0 1 7 6 . . . 7 6 ...... 7 6 c1 c2 cn 1 c0 7 4 5 and, therefore, it is determined by its first row. Toeplitz and circulant matrices with positive entries play an important role in numerical integration and define problems of “cyclic pursuit” in the n bugs problem. •
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MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 2.4 Eigenvalues, singular values, and induced norms 2 MATRIX CONCEPTS
2.4 Eigenvalues, singular values, and induced norms
We now prepare to determine general conditions that make a matrix convergent or semi-convergent. We start revisiting some notions and properties from matrix theory. We require the reader to be familiar with the notion of eigenvalue and of simple eigenvalue, that is, an eigenvalue with algebraic and geometric multiplicity2 n n equal to 1. The set of eigenvalues of a matrix A R ⇥ is called its spectrum and is denoted by spec(A) C. n n 2 T ⇢ The singular values of the matrix A R ⇥ are the positive square roots of the eigenvalues of A A. 2 We begin with a well-known property of the spectrum of a matrix.
Theorem 2.5 (Gerˇsgorin disks) Let A be an n n matrix. Then ⇥ n spec(A) z C z aii aij . ⇢ 2 k kC | | i 1,...,n j=1,j=i 2{[ } n X6 o Next, we review a few facts about normal matrices, their eigenvectors and their singular values.
n n Lemma 2.6 (Normal matrices) For a matrix A R ⇥ , the following statements are equivalent: 2 (i) A is normal; (ii) A has a complete orthonormal set of eigenvectors; and
3 (iii) A is unitarily similar to a diagonal matrix, that is, there exists a unitary matrix U such that U ⇤AU is diagonal.
Lemma 2.7 (Singular values of a normal matrix) If a normal matrix has eigenvalues 1,..., n , then its singular values are ,..., . { } {| 1| | n|} It is well known that real symmetric matrices are normal, are diagonalizable by orthogonal matrices, and have real eigenvalues. We conclude by defining the notion of induced norm of a matrix. For p N ,thep-induced norm of n n 2 [{1} A R ⇥ is 2 A = max Ax x =1 . k kp {k kp |k kp } One can see that n n
A 1 = max aij , A = max aij , k k j 1,...,n | | k k1 i 1,...,n | | 2{ } i=1 2{ } j=1 X X A = max is a singular value of A . k k2 { | }
2.5 Spectral radius and general criterion for convergent matrices
n n The spectral radius of a matrix A R ⇥ is 2 ⇢(A) = max spec(A) . {k kC | 2 } In other words, ⇢(A) is the radius of the smallest disk centered at the origin that contains the spectrum of A.
2The algebraic multiplicity of an eigenvalue is the multiplicity of the corresponding root of the characteristic equation. The geometric multiplicity of an eigenvalue is the number of linearly independent eigenvectors corresponding to the eigenvalue. The algebraic multiplicity is greater than or equal to the geometric multiplicity. 3 n n 1 AcomplexmatrixU C ⇥ is unitary if U = U ⇤. 2 5
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 3 BIRKHOFF–VON NEUMANN THEOREM
Lemma 2.8 (Induced norms and spectral radius) For any square matrix A and in any norm p 2 N , ⇢(A) A p. [{1} k k We will often deal with matrices with an eigenvalue equal to 1 and all other eigenvalues strictly inside the unit disk. Accordingly, we generalize the notion of spectral radius as follows. For a square matrix A with ⇢(A) = 1, we define the essential spectral radius
⇢ (A) = max spec(A) 1 . (4) ess {k kC | 2 \{ }}
Next, we provide a more general characterization of convergent and semi-convergent matrices. These two notions are characterized as follows.
Lemma 2.9 (Convergent and semi-convergent matrices) The square matrix A is convergent if and only if ⇢(A) < 1. Furthermore, A is semi-convergent if and only if the following three properties hold:
(i) ⇢(A) 1; (ii) ⇢ess(A) < 1,thatis,1 is an eigenvalue and 1 is the only eigenvalue on the unit circle; and (iii) the eigenvalue 1 is semisimple, that is, it has equal algebraic and geometric multiplicity (possibly larger than one).
In other words, A is semi-convergent if and only if there exists a nonsingular matrix T such that
I 0 A = T k T 1, 0 B (n k) (n k) where B R ⇥ is convergent, that is, ⇢(B) < 1. With this notation, we have ⇢ess(A)=⇢(B) and the algebraic2 and geometric multiplicity of the eigenvalue 1 is k. Consider now any example of MAS described by means of a row-stochastic matrix A. Because A is row- stochastic, it holds that A1n = 1n. Therefore, ⇢(A) is not strictly less than one and A is not convergent. Still A can be semi-convergent.
3 Birkho↵–von Neumann theorem
In this section, we step aside to provide an interesting result on doubly-stochastic matrices. First, we say k that the scalars µ1,...,µk are convex combination coe cients if µi 0, for i 1,...,k , and i=1 µi = 1. (Each row of a row-stochastic matrix contains convex combination coe cients.)2{ A convex} combination of vectors is a linear combination of the vectors with convex combination coe cients. A subset UPof a vector space V is convex if the convex combination of any two elements of U takes value in U. One can visualize convex sets as sets for which the segment joining any two points in the set is contained in the set. An interesting example of convex set is the set of stochastic matrices and the set of doubly stochastic matrices as is proven next.
Theorem 3.1 (Birkho↵–von Neumann) A square matrix is doubly stochastic if and only if it is a convex combination of permutation matrices.
Proof: Su ciency is clear. Let us establish the necessity. The strategy is to show that a matrix is an extreme point of the compact convex set of doubly stochastic matrices if and only if it is a permutation
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MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4 PERRON-FROBENIUS THEORY matrix. If this is true, then the result follows by noting that every point in a compact, convex set is a convex combination of the extreme points of the set. If A is a permutation matrix, then there is precisely one 1 entry in each row and column, and all other entries are 0. If A was not an extreme point, then it could be written as A = ↵1B + ↵2C, for 0 <↵1,↵2 < 1 and B, C doubly stochastic. In particular, whenever aij = 0, we have 0 = ↵1bij + ↵2cij,whichimpliesbij =0=cij. Since B and C are doubly stochastic, their row sums are 1 and hence nonzero entries must all be 1 and in the same positions as the nonzero entries of A. In other words, A = B = C, which shows that A is an extreme point. Finally, let us show that every extreme point is a permutation matrix. We proceed with the counter-positive, i.e., we show that if A is not a permutation matrix, then it cannot be an extreme point. Because A is not a permutation matrix, there must a row with at least two nonzero elements. Let a (0, 1) be one of ii2 2 them. Using again that A is stochastic, we deduce that there exists ai3i2 (0, 1), with i3 = i1. By the same reasoning, there exists a (0, 1), i = i . If this process is continued, after2 finitely many6 steps, there will i3i4 2 4 6 2 be a first time that an entry, aij is chosen that had already been chosen before. Consider the sequence of entries in A starting from the first time aij is chosen to the second time (without repeating the entry). Let ai0j0 be the smallest number in this sequence (all of them are di↵erent from zero, by construction). Let us n n now define a new matrix, B R ⇥ in the following way: 1 in the same position of A as the first entry (aij) 2 of the sequence; 1 in the same position of A as the second entry (aij) of the sequence; and so on. All the other entries are zero. By construction, all the rows and columns of B sum zero. Now, consider
A+ = A + ai j B, A = A ai j B. 0 0 0 0
Both these matrices are stochastic! This is because they are nonnegative (because of the way we choose ai0j0 ) 1 1 and their rows and columns sum 1 (because the rows and columns of B sum zero). Moreover, A = 2 A++ 2 A , and since A = A , it follows that A is not an extreme point. 6 +
4 Perron-Frobenius theory
Nonnegative matrices (and row-stochastic matrices in particular) are used in many fields, including economics, population models and biology, chemical networks, Markov chains, social networks, and power control in communications. The following is an additional example. (Note: in this section the notation A 0(reps.A> 0) means the entries of the matrix are a 0(resp.a > 0).) ij ij Example 4.1 (Economic sectors) Nonnegative matrices play an important role in the theory of input- output linear economic models proposed by Leontief. In particular, these models can be used to explain how component parts of an economy influence each other and fit together. Consider an economy with activity level x 0 in sector i 1,...,n . We can think of x being the output i 2{ } i or the commodity of a given sector of the economy. Given activity level xt at time t,inperiodt +1wehave
xt+1 = Axt with A 0. Here, a represents the units of good j required to produce good i and n a x is the output ij j=1 ij j of good i.4 P • Positive and nonnegative matrices have useful spectral properties. Perron-Frobenius theory helps characterize the spectral properties of these matrices. We refer to [3, Chapter 8] for a detailed treatment. We start with a useful auxiliary result.
4 In the Leontief model the amount aij represents the units of good i needed to produce good j.Thisisequivalenttoconsider the transpose with A in the previous description. 7
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4.1 Positive matrices 4 PERRON-FROBENIUS THEORY
Lemma 4.2 If the square matrix A is nonnegative, then, for any positive vector x, we have
1 n 1 n min aijxj ⇢(A) max aijxj i 1,...,n x i 1,...,n x 2{ } i j=1 2{ } i j=1 X X n n aij aij min xj ⇢(A) max xj j 1,...,n x j 1,...,n x 2{ } i=1 i 2{ } i=1 i X X In particular, this implies that, if ↵, 0 are such that ↵x Ax x, then ↵ ⇢(A) ,andthe corresponding strict inequality hold.
4.1 Positive matrices
In what follows, the first theorem amounts to the original Perron’s Theorem for positive matrices and the following theorems are the extension due to Frobenius for certain nonnegative matrices.
Theorem 4.3 (Perron-Frobenius for positive matrices) If the square matrix A is positive, then
(i) ⇢(A) > 0; (ii) ⇢(A) is an eigenvalue, it is simple, and ⇢(A) is strictly larger than the magnitude of any other eigenvalue; and
(iii) ⇢(A) has an eigenvector with positive components.
n n n n n Proof: We start by defining A =(aij ) for A R ⇥ and v =(vi ) for v R . k kC k kC i,j=1 2 k kC k kC i=1 2 Note that Av C A C v C (prove it!). Here, A B means that A B 0, i.e., all entries of A B are nonpositive.k k k k k k
(i) follows from using Lemma 4.2 with x = 1n. (ii) and (iii). Let spec(A) such that ⇢(A)= . Let x = 0 be an eigenvector of A corresponding to 2 k kC 6 the eigenvalue . Observe that Ax C = x C = C x C = ⇢(A) x C. If we were able to prove that Ax = A x , then an eigenvector-eigenvaluek k k k pairk isk givenk k by x andk k ⇢(A). k kC k kC k kC Define y = A x ⇢(A) x . Note that k kC k kC ⇢(A) x = x = x = Ax A x = A x k kC k kCk kC k kC k kC k kCk kC k kC (where we have used that A is positive), so that y 0. Let us show that y = 0. Assume not, i.e., y = 0. Note that, since A is positive and x = 0, then z = A x > 0. Note that 6 k kC 6 k kC 0 < Ay = A(A x ⇢(A) x )=Az ⇢(A)z, k kC k kC or, in other words, ⇢(A)z < Az. Using Lemma 4.2, this would imply that ⇢(A) <⇢(A), a contradiction. Therefore, y = A x ⇢(A) x = 0. Using (i), we have k kC k kC 1 x = (A x ) > 0 k kC ⇢(A) k kC
Therefore, we have proved that ⇢(A) is an eigenvalue and has an eigenvector x with positive components. k kC Note that any positive eigenvector v of A must correspond to ⇢(A). This is because, if Av = v,then 0, and using Lemma 4.2,wededuce ⇢(A) . To show that ⇢(A) is strictly larger than any other 8
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4.2 Nonnegative matrices 4 PERRON-FROBENIUS THEORY
eigenvalue, we build on the fact that if Av = v,with C = ⇢(A). This implies that v C and ⇢(A) are a positive eigenvector-eigenvalue pair. Then, it holds thatk k k k
⇢(A) v = v = a v a v = a v = ⇢(A) v k kkC k kkC k kp pkC k pkkCk pkC pkk pkC k kkC p p p X X X
The only way in which the inequality becomes equality is if all the complex vectors vk are on the same ray. i✓ Let ✓ R be the angle of this ray, then it holds that e v = v > 0. Consider an eigenvalue = ⇢(A). 2 k kC 6 By definition, C ⇢(A). Suppose C = ⇢(A) and let v be an eigenvector. By the fact mentioned above, k k i✓ k k there exists ✓ R such that w = e x>0. Since w is a positive eigenvector of A, it must be that = ⇢(A). 2 Finally, we show that ⇢(A) has geometric multiplicity one. Let w and z be two eigenvectors corresponding i✓1 i✓2 to ⇢(A). By the result above, there exist ✓1,✓2 R such that p = e w>0 and q = e z>0. Let 2 =mini 1,...,n qi/pi. Let r = q p. Notice that r 0 and at least one coordinate of r is zero, hence this vector2{ is not} positive. However, Ar = Aq Ap = ⇢(A)q ⇢(A)p = ⇢(A)r. Therefore, we have 1 r = ⇢(A) (Ar), which implies that, if r = 0, then r>0, a contradiction. 6 The algebraic multiplicity of ⇢(A) is also one. This can be proven as a consequence of Corollary 4.11.We will revisit the proof after that result.
The eigenvalue ⇢(A) is usually called the Perron-Frobenius eigenvalue, denoted pf . The normalized positive n eigenvector v (normalized meaning that i=1 vi = 1) is called the Perron-Frobenius eigenvector. P 4.2 Nonnegative matrices
Requiring the matrix to be strictly positive is a key assumption that limits the applicability of Theorem 4.3. In fact, for the economic example described in Example 4.1, we can not conclude anything unless A>0. If the matrix is only nonnegative, then the conclusions that can be reached are slightly weaker. However, it turns out that it is possible to obtain the similar results if we further require the nonnegative matrix to be irreducible.
n n Definition 4.4 (Irreducible matrix) A nonnegative matrix A R ⇥ is irreducible if, for any nontrivial partition J K of the index set 1,...,n , there exist j J and k2 K such that a =0. [ { } 2 2 jk 6 Remark 4.5 (Properties of irreducible matrices) An equivalent definition of irreducibility is given as n n follows. A matrix A R ⇥ is irreducible if it is not reducible, and is reducible if either: 2 (i) n = 1 and A = 0; or
n n T (ii) there exists a permutation matrix P R ⇥ and a number r 1,...,n 1 such that P AP is block upper triangular with diagonal blocks2 of dimensions r r and2 ({n r) (n } r). ⇥ ⇥ It is an immediate consequence that the property of irreducibility depends upon only the patterns of zeros and nonzero elements of the matrix. • Example 4.6 Consider the matrices
11 01 A = and A = 1 00 2 10 Justify that A is reducible and A is irreducible. 1 2 • 9
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4.2 Nonnegative matrices 4 PERRON-FROBENIUS THEORY
We can now weaken the assumption in Theorem 4.3 and obtain a comparable, but weaker, result for irreducible matrices.
Theorem 4.7 (Perron–Frobenius for irreducible matrices) If the square matrix A is nonnegative, then
(i) ⇢(A) 0; (ii) ⇢(A) is an eigenvalue, and (iii) ⇢(A) has an eigenvector with nonnegative components.
Furthermore, if A is irreducible, then
(i) ⇢(A) > 0; (ii) ⇢(A) is an eigenvalue, and it is simple; and (iii) ⇢(A) has an eigenvector with positive components.
In general, the spectral radius of a nonnegative irreducible matrix does not need to be the only eigenvalue of 01 maximum magnitude. For example, the matrix has eigenvalues 1, 1 . In other words, irreducible 10 { } matrices do indeed have weaker spectral properties than positive matrices. Therefore, it remains unclear which nonnegative matrices have the same properties as those stated for positive matrices in Theorem 4.3.
Definition 4.8 (Primitive matrix) A nonnegative square matrix A is primitive if there exists k N such that Ak is positive. 2 • It is easy to see that if a nonnegative square matrix is primitive, then it is irreducible. In particular, we have the following set of implications: A positive = A primitive = A irreducible = A nonnegative ) ) ) The implications can not be reversed. That is, one can find examples of matrices that are e.g. primitive but not positive, irreducible but not primitive, and nonnegative but not irreducible.
Example 4.9 Justify that the matrix A2 from Example 4.6 is not primitive. Consider the matrices 1100 011 1 1 0011 A3 = 101 and A4 = 2 3 2 2 1100 21103 600117 4 5 6 7 Justify that they are nonnegative, irreducible, and primitive. 4 5 • In the next set of lecture notes, we will provide a graph-theoretical characterization of primitive matrices; for now, we are finally in a position to sharpen the results of Theorem 4.7.
Theorem 4.10 (Perron–Frobenius for primitive matrices) If the nonnegative square matrix A is prim- itive, then
(i) ⇢(A) > 0; (ii) ⇢(A) is an eigenvalue, it is simple, and ⇢(A) is strictly larger than the magnitude of any other eigenvalue; and (iii) ⇢(A) has an eigenvector with positive components.
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MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4.3 Application to dynamical systems 4 PERRON-FROBENIUS THEORY
4.3 Application to dynamical systems
We conclude by noting the following convergence property for primitive, nonnegative matrices A, which has implications on the limiting behavior of the iteration x(k + 1) = Ax(k)=Akx as k + . 0 ! 1 1 Corollary 4.11 If the nonnegative square matrix A is primitive, then the matrix ⇢(A) A is semi-convergent. In fact,
1 t T lim (⇢(A) A) = vw , t !1 where v is the right Perron vector, w is the left Perron vector, and vT w =1.
1 Proof: The fact that ⇢(A) A is semi-convergent is an immediate corollary to Lemma 2.9 and Theo- rem 4.10. To determine the actual limit of the matrix, let L = vwT . Note that
LA = vwT A = ⇢(A)vwT = ⇢(A)L AL = AvwT = ⇢(A)vwT = ⇢(A)L Lt = vwT vwT = vwT = L ··· From this, it is not di cult to see that
(A ⇢(A)L)t = At ⇢(A)tL. (5) Also, note that any nonzero eigenvalue of A ⇢(A)L must be an eigenvalue of A, and moreover, ⇢(A) is not an eigenvalue of A ⇢(A)L. To show these facts, let µ = 0 be an eigenvalue of A ⇢(A)L.Then, 6 (A ⇢(A)L)x = µx, for some x = 0. Left multiplying by L and using the equalities above, we get 6 L(A ⇢(A)L)x = LAx ⇢(A)LLx = ⇢(A)Lx ⇢(A)Lx =0=µLx. Since µ = 0, it follows that Lx = 0, and hence Ax = µx. In addition, if µ = ⇢(A), then, because this eigenvalue is simple,6 we would have x = v for some = 0. But then ⇢(A)x =(A ⇢(A)L)x =(A ⇢(A)L) v = ⇢(A)v ⇢(A)v = 0, which is a contradiction6 with ⇢(A) > 0. From the above discussion, we deduce that ⇢(A ⇢(A)L) equals 0 or k C, for some k spec(A) ⇢(A) . 1 k k 2 \{ } In any case, ⇢(A ⇢(A)L) <⇢(A) or, equivalently, ⇢(⇢(A) A L) < 1. Using now (5), we get 1 t 1 t (⇢(A) A) = L +(⇢(A) A L) from which the result follows.
Lemma 4.12 For a primitive matrix A, the eigenvalue ⇢(A) has algebraic multiplicity equal to one.
1 t Proof: Since A primitive, we have that limt (⇢(A) A) = L, a rank one matrix. Suppose the multiplicity of ⇢(A)isk>1. Then, by the Shur triangularization!1 theorem, there exists a unitary matrix U such that A = U U ⇤, where has in the diagonal k values at ⇢(A) and others are strictly smaller. If k>1, we would have
lim U⇢(A)AU ⇤ = UBU⇤ t !1 with B having at least rank k, a contradiction. 11
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. 4.3 Application to dynamical systems 4 PERRON-FROBENIUS THEORY
Example 4.13 (Economic sectors–revisited) Recall the model described in Example 4.1 and further assume that A is primitive. Let pf be the Perron-Frobenius eigenvalue with right eigenvector v (normalized, T T i.e., 1n v = 1) and left eigenvector w such that v w = 1. Then, using Perron-Frobenius theory, we can deduce
(i) x (t + 1)/x (t), the growth factor in sector i over the period from t to t + 1, converges to as t . i i pf !1 t 1 t T T This is because limt pf x(t)=limt ( pf A) x(0) = vw x(0) = (w x(0))v.Sincex(t + 1) = Ax(t), we get !1 !1
t t t T T lim x(t + 1) = lim Ax(t)=A lim x(t)=A(w x(0))v = pf (w x(0))v t pf t pf t pf !1 !1 !1
n (ii) the distribution of economic activity (i.e.,x ˜ withx ˜i = xi/ i=1 xi) converges to v.
t T P This is because, again noting limt x(t)=(w x(0))v, one has !1 pf t x(t) lim x˜(t)= lim pf = v. t t t T !1 !1 pf x(t) 1n
(iii) asymptotically the economy exhibits balanced growth, by the factor pf , in each sector.
This is a consequence of (i) above. All these conclusions hold independently of the original economic activity, provided it is nonnegative and nonzero. • Example 4.14 (Roundtable example) Consider a roundtable example where five individuals sit around a table and each individual talks to the right and left neighbor to find out the average age of the group. Then, in compact form, the algorithm reads like
x1(t + 1) 1/21/40 01/4 x1(t) x (t + 1) 1/41/21/40 0 x (t) 0 2 1 0 1 0 2 1 x3(t + 1) = 01/41/21/40 x3(t) Bx (t + 1)C B 001/41/21/4C Bx (t)C B 4 C B C B 4 C Bx (t + 1)C B1/40 01/41/2C Bx (t)C B 5 C B C B 5 C @ A @ A @ A where the estimates are initialized by each individual with his/her own age. Note that the system matrix, let us call it F , is nonnegative, but not positive. It is doubly stochastic and irreducible (can you show it?). In addition,
3/81/41/16 1/16 1/4 1/43/81/41/16 1/16 0 1 F 2 = 1/16 1/43/81/41/16 B1/16 1/16 1/43/81/4 C B C B 1/41/16 1/16 1/43/8 C B C @ A and hence F is primitive. Since 1 is an eigenvalue of F and, using Gerˇsgorin disks, one can show that ⇢(F ) 1, 1 we deduce that ⇢(F ) = 1, with right eigenvector v = n 1n and left eigenvector w = 1n. Consequently, from Perron-Frobenius, we deduce
t 1 T lim F = 1n1n . t !1 n 12
MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo. REFERENCES REFERENCES
Consequently, the individuals’ age estimates asymptotically converge to
t 1 T 1 T lim x(t)= lim F x(0) = 1n1n x(0) = 1n x(0)1n, t t !1 !1 n n i.e., the average age of the overall group. • Exercise 4.15 (Flocking example–revisited) Can you mimic the arguments in Example 4.14 to figure out what happens in the flocking example? Note that the system matrix is not in general column stochastic, so adjust for that! •
References
[1] F. Bullo, J. Cort´es, and S. Mart´ınez, Distributed Control of Robotic Networks, ser. Applied Mathematics Series. Princeton University Press, 2009. [2] F. Bullo, Lectures on Network Systems. Version 0.96, 2018, with contributions by J. Cortes, F. Dorfler, and S. Martinez. [3] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University Press, 1985.
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MAE247 – Cooperative Control of Multi-Agent Systems Based on the class lecture notes of J. Cort´es,and F. Bullo.