Eigenvalues of Random Matrices over Finite Fields

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1 Eigenvalues of Random Matrices over Finite Fields Kent Morrison Department of Mathematics California Polytechnic State University San Luis Obispo, CA September 5, 999 Abstract We determine the limiting distribution of the number of eigenvalues of a random n n matrix over F as n. We show that the limit of this distribution is Poisson with mean. The main tool is a theorem proved here on asymptotic independence for events defined by conjugacy class data arising from distinct irreducible polynomials. The proof of this theorem uses the cycle index for matrices developed by Kung and Stong. Several examples are given to illustrate the use of the asymptotic independence theorem. These include previous results about derangements and about cyclic and semi-simple matrices. The Cycle Index for M n For a prime power let F denote the field with elements. Let M n be the space of n n matrices over F and let GL n be the group of invertible matrices. To define the cycle index for M n we recall the rational canonical form of a matrix see, for example, [8]. Each matrix α M n gives rise to a finite dimensional F [z]-module isomorphic to a direct sum k l i i= j= F [z]/ λ i,j i where,..., k are distinct monic irreducible polynomials; for each i, λ i, λ i,2 λ i,li is a partition of n i = j λ i,j. If α is n n, then n = i n i deg i = i,j λ i,j deg i. Let λ i denote the partition of n i given by the λ i,j and define λ i = n i. The conjugacy class of α in M n is determined by the data consisting of the finite list of distinct monic irreducible polynomials,..., k and the partitions λ,..., λ k. We introduce an indeterminate x,λ for each pair of a monic irreducible polynomial and a partition λ of some non-negative integer. The conjugacy class data for α is encoded into the product x,λ... x k,λ k. We can write this more economically if we define λ α to be the possibly empty partition corresponding to in the rational canonical form for α. We regard the empty partition as the 99 Mathematics Subject Classification. Primary 05A6, 5A33, 60C05; Secondary 5A8, 5A2 Key words and phrases. Asymptotic probability, cycle index.

2 only partition of 0. Set the corresponding indeterminates x, = for all. Then we can use x,λ α for the monomial corresponding to α. Following Kung [7]and Stong [0], we define the cycle index of M n to be + n= u n GL n α M n x,λ α. The first definition of the cycle index due to Kung used indeterminates x m,λ for m a positive integer m and λ a partition of m. But this definition did not distinguish among different irreducible polynomials of the same degree. Stong later modified the definition of the cycle index to be the one used in this paper. Kung and Stong proved the following factorization holds. Theorem + n= u n GL n α M n x,λ α = λ x,λ u λ deg c λ where c λ is the order of the group of module automorphisms of the F [z]-module j F [z]/ λ j. Proof On the left side α M n x,λ α is the sum of all monomials corresponding to all n n matrices. Split this sum according to the conjugacy classes. Let,..., k and λ,..., λ k be the data of one conjugacy class. The monomial x i,λ i occurs with the coefficient eual to the size of the conjugacy class, namely GL n / G, where G is the centralizer of an element in the conjugacy class. But G is isomorphic to the the subgroup of automorphisms of the F [z]- module i j F [z]/ λ i,j i. An automorphism preserves the the i -primary component and so G = c i λ i. Then x GL n,λ α has the monomial x i,λ i occurring with coefficient α c λ... c k λ k which is exactly how the same monomial appears in the right hand side. In dealing with the cycle index for M n we have found Fulman s [4] notation and presentation most useful. There are several basic tools necessary for working with the cycle index. We give them as a seuence of lemmas as in [4]. First, from Kung [7] we get a formula for c λ contained in the next lemma. See that paper for the proof. Lemma 2 Let λ be the partition of n given by λ λ 2 λ k. Let m i be the number of parts of size i. Let d i = m + 2m im i + im i+ + + m n. Then c λ = n m i d i deg d i j deg. i= j= Note In the Ferrers diagram of λ consisting of λ i boxes in row i, one can see that d i is the number of boxes in the first i columns. 2

3 Lemma 3 Stong [0] Fix a monic irreducible. Then λ u λ deg c λ = udeg. r deg Proof It is sufficient to prove this when deg = because of the way c λ depends on the degree of. Furthermore, since c λ just depends on the degree, it is sufficient to prove it for z = z. Now we are dealing with u λ c λ which we split into an outer sum over n and an inner sum over the partitions of n λ λ =n u λ λ c λ = + n λ =n u n c λ. c λ To evaluate the inner sum we note that GL n λ =n is the number of nilpotent matrices of size n n. Fine and Herstein [3] proved that the number of n n nilpotent matrices is nn. Therefore, c λ = n.... n Next we use a formula from Hardy and Wright [6, Theorem 349, p. 280]. For a <, y <, the coefficient of a n in the infinite product is ay r y n y y 2... y n. With this formula let a = u and y = to see that the un coefficient of is u r n... n. A key lemma for evaluating asymptotic probabilities is found in Wilf [, Lemma B, p. 44] or Fulman [4]. For a power series fu = n a nu n, we let [[u n ]]fu mean the coefficient of u n. Lemma 4 If f is analytic at u = 0 and the power series for f converges at u =, then lim n [[un ]] fu u = f. 3

4 Proof Multiply fu = n 0 a nu n by u = + u + + u n The coefficient of u n is n k=0 a k. Now let n go to infinity. Lemma 5 udeg r deg = u r Proof Let r =. Then udeg deg = u k deg k 0 Expanding this product we use uniue factorization of polynomials in F [z] to see that the coefficient of u n is n because the coefficient in uestion is the number of monic polynomials of degree n. Thus, u k deg = u n n = u. n 0 Now replace u by u r. k 0 Lemma 6 λ u λ deg c λ = r 0 u r. Proof Follows easily from the previous lemmas. 2 Asymptotic Probability We will consider a subset A n M n as an event, although strictly speaking it is a family of events parameterized by n. For each n we have the event A M n contained in the finite probability space M n. Define the asymptotic probability of A as ˆPA := lim n A M n, M n if this limit exists. We are particularly interested in events that are defined by conjugacy class data. Let L be any subset of the set of all partitions of all non-negative integers and let ψ be a monic irreducible polynomial over F. Define the event Eψ, L to be the set of suare matrices {α λ ψ α L}. For example, the invertible matrices are Eψ, L where ψz = z and L is the singleton set consisting of the empty partition; in this case, ˆPEψ, L = r. Theorem 7 ˆPEψ, L = r deg ψ c ψ λ λ L 4

5 Proof Beginning with the cycle index and its factorization we use the lemmas above. GL n ˆPEψ, L = lim n M n [[un ]] u λ deg u λ deg ψ c λ c ψ λ ψ λ λ L = u λ deg r lim n [[un λ c ]] λ u λ deg ψ u λ deg ψ c λ ψ λ c ψ λ λ L = r 0 u r lim n [[un ]] r u λ deg ψ udeg ψ c ψ λ r deg ψ λ L = u r lim n [[un ]] r u u udeg ψ c ψ λ r deg ψ λ L r = r = r deg ψ c ψ λ λ L r deg ψ λ L c ψ λ λ deg ψ Theorem 8 Asymptotic Coprime Independence If ψ and ψ 2 are distinct monic irreducible polynomials over F and L and L 2 are subsets of all partitions of non-negative integers, then Proof ˆPEψ, L Eψ 2, L 2 GL n = lim n M n [[un ]] = r lim n [[un ]] r = lim n [[un ]] = r deg ψ ˆPEψ, L Eψ 2, L 2 = ˆPEψ, L ˆPEψ 2, L 2 ψ,ψ 2 u c λ λ λ u λ deg ψ c ψ λ λ deg u λ deg λ c λ λ L λ u λ deg ψ 2 c ψ λ u λ deg ψ c ψ λ u λ deg ψ2 c ψ λ λ L 2 u λ deg ψ c ψ λ λ L r 0 u r udeg ψ r deg ψ udeg ψ 2 r deg ψ 2 r deg ψ 2 c ψ λ c ψ2 λ λ L λ L 2 u λ deg ψ c ψ λ λ L u λ deg ψ2 c ψ2 λ λ L 2 u λ deg ψ2 c ψ2 λ λ L 2 = ˆPEψ, L ˆPEψ 2, L 2. 5

6 Since ˆP is not a probability measure on a sample space, we do not have countable additivity of ˆP. It is countable additivity that implies that the probability of a countable intersection of independent events is the product of the probabilities of the events. In light of that it is notable that we do have such a result for a countable intersection of events of the form Eψ i, L i. This corollary allows us to consider asymptotic probabilities for events defined by conditions on each monic, irreducible polynomial in F [z]. Corollary 9 For distinct, monic, irreducible polynomials ψ i, and arbitrary subsets of partitions L i, i =, 2,... ˆP Eψ i, L i = ˆPEψ i, L i. i= Proof The proof follows that of the theorem. In order to consider probability uestions involving invertible matrices, rather than all matrices, the next theorem is uite useful, because it shows that the asymptotic conditional probability of an event conditioned on invertibility is typically the same as the asymptotic unconditioned probability. For events A and A 2, we define the asymptotic conditional probability of A given A 2 as A A 2 M n PA A 2 M n ˆPA A 2 := lim = lim n A 2 M n n PA 2 M n Theorem 0 Let {ψ i } i I be a finite or countable set of distinct, monic, irreducible polynomials in F [z], none of which is eual to z. Let L i, i I, be subsets of partitions and let E i = Eψ i, L i. Then ˆP E i invertible = ˆP E i i i i= Proof A matrix α is invertible if λ z α =. Therefore, the subset of invertible matrices is Ez, { } which we will denote by E 0. ˆP P i E i invertible = lim E i E 0 M n n PE 0 M n i = lim n P i E i E 0 M n lim n PE 0 M n ˆP i = E iˆpe 0 ˆPE 0 = ˆP i E i Let X be a real or integer valued function on the union of the M n. We will refer to such an X as a random variable although it is actually a family of random variables, one for each discrete probability space M n. An important example, to be treated in detail in section 4, is the dimension of the kernel. Now for a random variable X taking values in N = {0,, 2,... }. we consider the asymptotic probabilities ˆPX = k for each k N. These may or may not define a probability distribution on N because of the limiting process. A trivial example of not being a probability distribution is given by X being the size of the matrix. In this case ˆPX = k = 0 for 6

7 all k. But for random variables expressed in terms of conjugacy class data we have the following theorem. Theorem If f is a function from partitions to N and X is the random variable defined by Xα = fλ α for a fixed monic, irreducible polynomial, then ˆPX = k =. k 0 Proof We let L k = f k and then E, L k is the event that X = k. By Theorem 7 ˆPX = k = ˆPE, L k = r deg c λ. λ L k Summing over k gives us ˆPX = k = k 0 r deg k 0 c λ. λ L k Now the sum k 0 λ L k c λ is the sum over all partitions. By Lemma 3 λ c λ = r deg. 3 The Number of Eigenvalues For each a F define the random variable X a to be the dimension of the a-eigenspace, that is, X a α = dim kerα ai. Thus, X a is the geometric multiplicity of a as an eigenvalue. This theorem gives the asymptotic distribution of X a. Notice that the most likely multiplicity is. Theorem 2 For k, ˆPX a = k = k 2... k 2 r For k = 0 ˆPX a = 0 = r Proof It is enough to show this for a = 0 since X 0 α ai = X a α. First we count the linear maps on F n with k-dimensional kernel. The number of possible subspaces of dimension k to serve as the kernel is n n... n k k k... k k This is a Gaussian binomial coefficient. Having chosen a subspace to be the kernel let v,..., v n k be complementary linearly independent vectors. They must be mapped to linearly independent 7

8 vectors. Now v can be mapped to any of n vectors, v 2 to any of n vectors, and so on, ending with v n k mapped to any of n n k vectors. Thus, the number of linear maps with k-dimensional kernel is n n... n k k k... k k n... n n k Dividing this by n2 we get the probability that an n n matrix has k-dimensional kernel, which is k 2... k 2 n i= i n i=n k+ i Now let n. Since X a is a random variable whose value on the matrix α only depends on the partition λ z a α, it follows from Theorem that the asymptotic probabilities ρ k := ˆPX a = k actually define a discrete probability distribution on the non-negative integers k = 0,, 2,.... However, this special case is euivalent to an identity of Euler. For a proof see Andrews [, 2.2.9, p. 2] and substitute / for. Corollary 3 Let L k be the set of partitions having exactly k parts and let be a monic irreducible polynomial of degree m. Then c λ = km m 2... km 2. λ L k Proof First, for deg = we have z = z a for some a F. Then λ α L k iff X a α = k. Now use the previous theorem. For the higher degree case just replace by m. We define ÊX a to be the limit as n of the expected value of X a on M n. The next theorem justifies an interchange of limits. Theorem 4 ÊX a = k 0 kρ k. Proof Let P n and E n denote probability and expected value on M n. Then ÊX a = lim E nx a n = lim kp n X a = k n k 0 We use the Dominated Convergence Theorem on the measure space {0,, 2,... } with counting measure. For each n the kp n X a = k define an integrable function on this measure space. From Theorem 2 we see that k k kp n X a = k < 2... k 2. 8

9 The right side is the kth term of a convergent series by the Ratio Test, and so Dominated Convergence implies lim kp n X a = k = lim kp nx a = k n n k 0 k 0 = kˆpx a = k k 0 = k 0 kρ k. Define the random variable X = a F X a, which counts the number of eigenvalues with multiplicity in F. Then ÊX is the asymptotic expected number of eigenvalues. Proposition 5 The asymptotic expected number of eigenvalues of a matrix over F is k k r 2... k 2. k Proof ÊX = Ê a F = ÊX 0, since the X a are identically distributed. Therefore, ÊX = k 0 kρ k. In order to describe the asymptotic distribution of the number of eigenvalues we make use of the probability generating function for the random variable X defined by f X t = ˆPX = kt k. We also have the probability generating functions for each of the X a given by f Xa t = k=0 ρ k t k = k=0 ˆPX a = kt k. Because of the asymptotic independence of the X a for a F we see that f X t = a F f Xa t, k=0 but the X a are also identically distributed. Therefore f X t = ρ k t k. k=0 Theorem 6 The asymptotic probability that a matrix over F has k eigenvalues in F counted with multiplicity is given by the coefficient of t k in the power series ρ k t k. k=0 9

10 In other words, this asymptotic probability is k +...k =k ρ k ρ k2... ρ k. The probability of no eigenvalue is the constant term of f X, which is ρ 0 = the probability of exactly one eigenvalue is the t-coefficient of f X, which is ρ 0 ρ = r. 2 r, and Example For = 2 the values of the ρ k are approximately ρ ρ ρ ρ For k 4, ρ k 0. The expected dimension of the null space is ÊX 0 = k kρ k 0.85 and the expected number of eigenvalues is 2ÊX 0.7. The distribution of the number of eigenvalues can be seen in the coefficients of the probability generating function f X t = ρ ρ 0 ρ t + 2ρ 0 ρ 2 + ρ 2 t 2 + 2ρ 0 ρ 3 + 2ρ ρ 2 t t t t t About 33% of the time a large binary matrix has just one eigenvalue and about 40% of the time it has two eigenvalues. The Maple code below simulates the sampling of 000 random matrices of size 5 over F 2 and finds the distribution of the dimensions of the null space. n:=5: # size of matrices num:=000: # number of random matrices A:=array..n,..n: C:=array0..n,[se0,i=0..n]: # Ck is the number of matrices with k-dimensional null space for kk to num do for i to n do for j to n do A[i,j]:=rand2: od; od; dim := nopsnullspacea mod 2: C[dim]:=C[dim]+: od: entriesc; In one run there were these results, which agree with the theoretical values for ρ k. dim ker proportion

11 The sample average is Theorem 7 As the distribution of X the number of eigenvalues in the base field approaches a Poisson distribution of mean. Proof The probability generating function for the Poisson distribution is e k 0 tk k! = et. By the Continuity Theorem see [2, p. 280] it will suffice to show that lim f X t = e t pointwise for 0 t. Now, Recall that ρ k = log f X t = log f X0 t = log f X0 t = log k 0 ρ k t k. k ρ 2... k 2 0 and ρ 0 =. Therefore, r log f X t = log ρ 0 + log k 2 tk. k From Corollary 23 we see that lim log ρ 0 =. The second term is log + k 2 t t Expand using the series for log + x and note that as the only term that does not go to 0 faster than / is t. Thus, the limit of the second term is 2 lim 2 t = t.. Hence lim log f X t = + t and lim f X t = e t. Example 2 Using Maple a simulation of 500 random matrices of size 5 by 5 over the field with 3 elements gave the following distribution of the number of eigenvalues. For comparison a Poisson distribution of mean is shown. # of eigenvalues proportion Poisson The random variable X counts the number of blocks in the rational canonical form associated to the linear monic polynomials. Euivalently, it counts the sum of the dimensions of the eigenspaces of the eigenvalues in the base field. Next we consider the eigenspaces associated to eigenvalues in extension fields. Let be a monic irreducible polynomial of degree m. Define the random variable V to be the number of parts of the partition λ. Thus V counts the number of blocks in the rational canonical form associated to. Proposition 8 The asymptotic probability that V = k is ˆPV = k = km m 2... km 2 rm.

12 Proof The proof follows from Theorem 7 and Corollary 3. Define V m = deg =m V. Notice that V is X above. Because the V are asymptotically independent, the asymptotic probability generating function f Vm t = f V t, where f V t = deg =m ˆPV = kt k. For all irreducible of degree m the functions f V t are identical and so k=0 f Vm t = f V t ν m, where is any fixed irreducible monic polynomial of degree m. Theorem 9 As the distribution of V m approaches a Poisson distribution of mean /m. Proof We have log f Vm t = ν m log f V t. Recall that ν m = m /m + O m/2. We can use the proof of Theorem 7 with m in place of to show that lim m log f V t = t. Easy estimates show that lim ν m log f V t = t. m Hence, as, f Vm t e m t, which is the the probability generating function of a Poisson distribution with mean /m. 4 Other Applications and Examples In this section we present results due to Stong and Fulman that illustrate the use of the results on asymptotic independence to find some interesting asymptotic probabilities. Proposition 20 For a F the asymptotic probability that a is not an eigenvalue is. r Proof Use Theorem 7 with ψz = z a and let L the singleton set containing the empty partition. Then λ L c ψ λ =. A derangement is a permutation with no fixed points. A linear map from F n to itself always fixes 0, so we define a linear derangement as an invertible linear map that has no non-zero fixed vectors. Proposition 2 The asymptotic probability that an invertible linear map is a linear derangement is. r 2

13 Proof A linear map fixes a non-zero vector precisely when it has for an eigenvalue, and the proposition follows from the previous theorem and proposition. This probability that an invertible linear map does not fix a non-zero vector is the same as the probability that any linear map does not fix a non-zero vector, which is the same as the probability of being invertible. Now we consider the natural action of GL n on projective space P n F. Define a projective derangement to be an element of GL n that acts on P n F without fixed points. Since points in P n F are lines in F n, the fixed points of α acting on P n F are the lines lying in eigenspaces of α. Proposition 22 The asymptotic probability that an invertible linear map is a projective derangement is. r Proof To be a derangement means that there are no eigenvalues in the base field F. Thus, the asymptotic probability is ˆPE, where E = Ez a, L a F \{0} and L is the singleton containing the empty partition. By the theorem on coprime independence ˆPE = ˆPEz a, L a F \{0} From Proposition 20 ˆPEz a, L =, and so r ˆPE = r. Recall that the probability that a permutation on n letters is a derangement approaches /e as n goes to infinity. For projective derangements we have to let the matrix size and the field size go to infinity. Corollary 23 The limit as of the asymptotic probability that an invertible linear map is a projective derangement is /e. That is, lim r = e. Proposition 24 Let a be algebraic of degree m over F.Then the asymptotic probability that a is not an eigenvalue of a suare matrix is. rm Proof The event we need is Eψ, L where ψ is the monic irreducible polynomial of degree m having a as a root and L is the singleton containing the empty partition. By Theorem 7 ˆPEψ, L = r deg ψ c ψ λ. Now λ L c ψ λ = and deg ψ = m. λ L 3

14 Proposition 25 Let be a monic, irreducible polynomial of degree m. Then the asymptotic probability that is a factor of the characteristic polynomial of a suare matrix is. rm Proof This is the complementary probability to that of the previous proposition. Remarks Contrast this result with the probability that divides a random monic polynomial of degree n as n. For n larger than m = deg this probability is / m. In this sense the characteristic polynomial is not distributed uniformly across the monic polynomials. On the other hand, Hansen and Schmutz [5] have shown that if we disregard the small factors, then the characteristic polynomial of a random matrix is random, but the meaning of small is relative to the size of the matrix, growing proportionally to the logarithm of the size of the matrix. We define α M n to be semi-simple if the associated F [z]-module is semi-simple, which means that it is isomorphic to a direct sum F [z]/ i i where the i are irreducible but not necessarily distinct. This property is also euivalent to being diagonalizable over the algebraic closure of F. In terms of the conjugacy class data, α being semi-simple is euivalent to λ α being the empty partition or consisting only of s for all monic irreducible. When the associated module is cyclic we say that α is cyclic. In that case the associated F [z]-module is isomorphic to F [z]/ e i i i where the i are distinct and irreducible. Euivalently, the minimal polynomial is the same as the characteristic polynomial of α. Also euivalent to α being cyclic is that it is regular when considered as an element of GL n F, where an element of an algebraic group is regular if its centralizer has minimal dimension. In terms of the conjugacy class data, α being cyclic is euivalent to λ α being empty or consisting of just one part for all. Proposition 26 The asymptotic probability that a matrix is semi-simple and cyclic euivalently, that the characteristic polynomial is suare-free is r. = r 2 Proof Let L = {, } be the subset containing just the empty partition and the uniue partition of. The set of matrices that are both semi-simple and cyclic is the intersection of E, L as ranges over the monic irreducible polynomials. Now λ L c λ = + and so deg ˆPE, L = r deg + deg r deg deg + deg = r 2 r deg. 4

15 It follows that ˆPsemi-simple and cyclic = ˆPE, L = = r 2 = r 2 = r 2 r deg r r deg r. Here Lemma 5 is used to go from the third to the fourth line. Consider the next proposition in light of the fact that the asymptotic probability is that a random polynomial is suare-free. Proposition 27 The asymptotic probability that an invertible matrix is semi-simple and cyclic euivalently, that the characteristic polynomial is suare-free is. Proof Let L = {, } as in the previous proposition. By Theorem 0 the asymptotic probability that an invertible matrix is semi-simple and cyclic is ˆPsemi-simple and cyclic invertible = ˆP E, L z = ˆPE, L z ˆPE, L = ˆPEz, L = ˆPsemi-simple and cyclic + r = + =. Remarks Fulman also finds the asymptotic probabilities that a matrix is semi-simple, that an invertible matrix is semi-simple, that a matrix is cyclic, and that an invertible matrix is cyclic. The calculation of the probability that a matrix is semi-simple uses Gordon s generalization of the Rogers-Ramanujan identities. There is earlier related work of Neumann and Praeger [9] on these probabilities. 5

16 Notice that the probabilites of being cyclic and semi-simple go to as, which is to be expected since these properties are generic for the Zariski topology in the algebraic varieties of n n matrices and invertible n n matrices over the algebraic closure F. References [] G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 976. [2] W. Feller, An Introduction to Probability Theory and Its Applications. Vol. I., John Wiley & Sons, Inc., New York-London-Sydney, second ed., 968. [3] N. J. Fine and I. N. Herstein, On the number of nilpotent matrices with coefficients in a finite field, Illinois J. Math., 5 958, pp [4] J. Fulman, Cycle indices for the finite classical groups, J. Group Theory, 2 999, pp [5] J. C. Hansen and E. Schmutz, How random is the characteristic polynomial of a random matrix, Math. Proc. Camb. Phil. Soc., 4 993, pp [6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press, New York, fifth ed., 979. [7] J. Kung, The cycle structure of a linear transformation over a finite field, Linear Algebra Appl., 36 98, pp [8] S. Lang, Algebra, Addison-Wesley, Reading, Mass., third ed., 993. [9] P. M. Neumann and C. E. Praeger, Cyclic matrices over finite fields, J. London Math. Soc., , pp [0] R. Stong, Some asymptotic results on finite vector spaces, Adv. Appl. Math., 9 988, pp [] H. S. Wilf, Generatingfunctionology, Academic Press, second ed.,