Proc. Indian Acad. Sci. (Math. Sci.), Vol. 105, No. 1: August 1995, pp. 269-271. �9 Printed in India.

Irrationality of linear combinations of eigenvectors

ANTHONY MANNING Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK

MS received 25 November 1994

A~traet . A given n x n matrix of rational numbers acts on C" and on Q'. We assume that its characteristic polynomial is irreducible and compare a basis of eigenvectors for C" with the standard basis for Q". Subject to a hypothesis on the Galois group we prove that vectors from these two bases are as independent of each other as possible.

Keywords. Irrationality; Galois group; eigenvectors.

A square matrix Ae GL(n, Q) can be considered as acting on Q" and on Qn| C = C n. The action on C ~ is best understood in terms of eigenvectors and that on Q" in terms of the standard basis e 1 . . . . . e, where e i = (6o)~= 1. We shall study the possibility of linear dependence (over C) between vectors from these two bases.

An eigenvector corresponding to an irrational eigenvalue clearly cannot lie in Q~. But can it lie in V | C where V is some codimension one subspace of Q~? How many of the coordinates of an eigenvector can be rational? And could a non-zero C-linear combination o f t eigenvectors lie in V| C where V is some codimension r subspace of Qn? Because we can work with A conjugated by a change of basis matrix in GL(n, Q) it is sufficient to consider these questions for subspaces spanned by vectors of the standard basis of Qn.

To avoid rational eigenvalues let us assume that the characteristic polynomial x(A) is irreducible over Q. Since z(A) is separable there are n distinct eigenvalues and A is diagonalizable. Moreover, there is no A-invariant subspace of Q~. Now if there was an A-invariant subspace U then eigenvectors in U | C would be linearly dependent over C on vectors that form a Q-basis for U, and we have avoided this type of possibility of linear dependence by the hypothesis that z(A) is irreducible.

Let F denote the splitting field extension of z(A) over Q and F denote the Galois group of this extension. Then F acts on the set of roots of ;~(A). This action is transitive [1, p. 66]. When 1 ~< r < n we call the action r-homogeneous if, for any two subsets consisting of r roots, there is an element of F that takes the elements of the first set to those of the second. Crrtainly the action of F is 1-homogeneous. It is r-homogeneous if and only if it is (n - r)-homogeneous. If F is the symmetric or alternating group on n symbols then the action is r-homogeneous for each r.

Theorem. Suppose that the characteristic polynomial z( A ) of A e G L(n, Q) is irreducible, so that the eigenvalues ~t 1 . . . . . ~t n of A are distinct. Consider a matrix Be GL(n, C) whose i-th row is a left eigenvector of A corresponding to the eigenvalue ati, 1 <~ i <<. n. Let F denote the Galois group of the splitting field extension F:.Q of z(A), and f ix t <~ r < n. If, for each q with 1 <~ q <~ r, the action of F on {~1 . . . . . ct,} is q-homogeneous then every r x r minor of B has non-zero determinant.

269

270 Anthony M annin,q

Remark 1. The determinant of an r x r minor of B is equal (at least up to sign) to the determinant of a matrix obtained from B by replacing the n - r rows not in that minor by the n - r vectors of the s tandard basis that do not correspond to any of the r columns in the minor. Thus the theorem asserts that any set of r eigenvectors and n - r vectors from the s tandard basis is independent. (Independence over F is equivalent to independence over C since both are equivalent to the vanishing of the determinant.) Two corollaries follow immediately.

C O R O L L A R Y 1

I f F is the symmetric or alternating group on n symbols then any set of n vectors taken from among the standard basis vectors and eigenvectors corresponding to different eigenvalues is independent over C.

C O R O L L A R Y 2

An eigenvector of A cannot lie in V | C when V is a codimension one subspace of Q".

Remark 2. If x = (x i . . . . . x,) e C" has x i . . . . . x, + i e Q then x = f l + ~ =, + 2 Xj ej e V | C for the codimension r subspace V of Q" spanned by f , ,e ,+ 2 . . . . . e n where f~:=Y,~+__]xjejeQ". Thus Corol lary 2 implies that no eigenvector can have two coordinates rational; the conclusion of the Theorem implies that a C-linear combina- tion of r eigenvectors can never have r + 1 coordinates rational. However, the precise number of coordinates that are rational can change if we change the basis of Q ' .

Remark 3. If

0 0 0 - 1

1 0 0 0 A =

0 1 0 1

0 0 1 0

then g(A) = x 4 - x 2 + 1 is irreducible and A has eigenvalues _ ct, • 0t- 1 for ~t a square

root of (1 + ix/~)/2. Then Galois g roup of x(A) is the Klein four group, which is not 2-homogeneous. So this case does not satisfy the hypotheses of the theorem if r = 2. A has left eigenvectors v I = (1, ~t, ~t 2, ~t 3 ) and v 2 = (1, - ~t, ~t 2, - at a ) cor responding to the eigenvalues __+ ~t. But then v 1 + v 2 = 2el + 2ct2e3 so that these four vectors are linearly dependent and the conclusion of the theorem is not satisfied. Thus some hypothesis on the Galois g roup is needed.

Remark 4. The theorem arose from work on hyperbolic total automorphisms. Here A is assumed to have only integer entries and de t (A)= +__ 1. Then A induces an au tomorphism ,4 of the quotient g roup R"/Z n, which is the n-dimensional torus T n. A vector subspace of R n that has a basis in Q" or Z" corresponds to a lower-dimensional torus in T*. If A has no eigenvalue of modulus 1 the toral au tomorph i sm ,4 is called hyperbolic. A hyperbolic A has elaborate dynamical properties: on the one hand, for some xe T" the orbit {,4kx:keZ} is dense in T", on the other hand the periodic (i.e. finite) orbits are the orbits of rational points (i.e. points of Q n / z ' ) and these form a dense subset of T*. (See Theorems 3.3 and 6.2 of [2] or 1.11 of 1-3].) Study of these dynamical

Irrationality o f combinations of eigenvectors 271

propert ies uses R " = E~t~E" where E ~= {veR":Akv-~O as k ~ oo} and E ~ = {veR ' : A k v ~ O as k ~ - ~ } . O u r theorem gives algebraic condi t ions under which the projections of E ~ and E ~ to T" are in general posi t ion with respect to the lower- dimensional tori. Let X, denote the characteris t ic po lynomia l of the a u t o m o r p h i s m induced by ,4 on the homology g roup H, (T" ) . Then the roots of Zr are products of r distinct roots o fz (A ). If Z, is irreducible then the act ion of F is r -homogeneous , which helps in checking the hypothesis of the theorem.

Proof of Theorem. We work in F" where (A - 0t I I ) . . . (A - ~qI) = 0 and each A - ~t,l has nullity one. Fo r each j, choose a left e igenvector vjeF" cor responding to ~j.

Any element ~ of the Galois g roup F is a field i somorphism o : F ~ F that leaves Q fixed pointwise. ~r induces a pe rmuta t ion of {~1 . . . . . 0~,} and we shall write ~(~j) = %(~,~j).

Now t~ induces a Q-l inear m a p O:F"-~F". U p to mult ipl icat ion by constants , permutes the eigenvectors of A because O(vj)A = O(vjA) = O(ctjvj) = ~(0tj)t~(vj) so that

O(vj) = c(~,j)v~(o)(j) for some non-zero c(~ , j )eF. N o w suppose, if possible, that r vectors f rom {vl . . . . . v, } and n - r f rom {e: . . . . . e. }

in F" are linearly dependent over F and tha t r is the least n u m b e r for which this is true. By Remark 1 it suffices to find a cont radic t ion to the existence of such vectors. By renumber ing if necessary we can assume tha t the vectors are v t , . . . , v,, e,+ 1,- .- , e,. By the dependence there are fll . . . . . f l .~F, not all zero, with

f l jv j= ~ fljej. (1) j = l j = r + l

Since r is the least possible, flj ~ 0 for 1 ~<j ~< r. Since the Galois g roup F is r -homogeneous we can, for k = 0, 1 . . . . . n - r, find a ~ e F

for which the pe rmu ta t i on 7r(~k) maps {1 . . . . . r} to {k + 1 . . . . . k + r}. Apply each tT k to (1). This gives

6k(flj)C(tTk,j)l)~t(e\)(j) = ~ (~k(flj)ej, O<~k < ~ n - r . (2) j = l j = r + l

The n - r + l vectors on the left hand sides of (2) all lie in the (n - r)-dimensional subspace of F" spanned by e, + 1 . . . . . e. so they are linearly dependent over F. Thus, for some k < ~ n - r , F.~=l~k(fli)c(~k,j)v.(~k)o ) is a linear combina t ion of Y~=l~m(flj) c(trm,j )v.(r ~, 0 ~ m < k. Now ~k(v j ) = c(~k,j )v.tr = c(~k,j )v,+ k w h e n j = (lt(tT,))- 1 (r + k). Fo r this value o f j it is tTk(flj)C(~k,j) tha t is the (non-zero) coefficient of v,+, and so v,+ k is a linear combina t ion of v I . . . . . v ,+,_ 1, which contradicts the independence of the eigenvectors, and so completes the proof.

R e f e r e n c e s

[!] Garling D, A course in Galois theory (Cambridge: University Press)(1986) [2] Smale S, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967) 747-817 [3] Walters P, An introduction to eroodic theory (New York: Springer) (1982)