= W(XI - A) = W(Ci(Ai - A)),

Transcription

1 proceedings of the american mathematical society Volume 14, Number 2, October 1988 POLYNOMIALS AND NUMERICAL RANGES CHI-KWONG LI (Communicated by Louis J. Ratliff, Jr.) ABSTRACT. Let A be an n x n complex matrix. For 1 < fc < n we study the inclusion relation for the following polynomial sets related to the matrix A. (a) The classical numerical range of the fcth compound of the matrix XI-A. (b) The fcth decomposable numerical range of the matrix XI A. (c) The convex hull of the set of all monic polynomials of degree fc that divide the characteristic polynomial of A. Moreover, we give an example showing that the set described in (a) is not convex in general. This settles a question raised by C. Johnson. Results and examples. Let Cnxn be the set of all n x n complex matrices. For A G CnXn> its (classical) numerical range is the set W(A) = {xax* : x G C" and xx* = 1}. Suppose A is hermitian and its characteristic polynomial p^(a) = det(af A) has roots Ai > > An. For 1 < k < n, we denote by Int(p,i(A); k) the set of all monic polynomials q(x) of degree fc that have k real roots pi > > p-k satisfying Xl>ßl and p.k-i+1 > ^n-i+i for t = 1,...,fc. Let Pk(XI A) be the convex hull of the set Johnson [3] proved the interesting result that (1) Int(pA(A);n - 1) = W(adj(Af - A)) = Pn_, (Af - A). For B G Cnxn and 1 < fc < n, let Ck(B) be the fcth compound of B (see [4] for definitions and properties). Since W(Bl) = W(B) and Cn-\(B) is unitarily similar to adj(s)', we may replace 'W(adj(Ai - A))" by 'W(C _i(ai - A))" in (1). Clearly, Int(p^(A);l) = {A-p: A, > p > A } = {A-p:pGW(A)} = W(XI - A) = W(Ci(Ai - A)), Received by the editors August 1, 1987 and, in revised form, November 2, Mathematics Subject Classification (1985 Revision). Primary 15A6, 15A69, 26C1. Key words and phrases. Numerical range, decomposable numerical range, characteristic polynomial. Research supported by NSF grant DMS American Mathematical Society /88 $1. + $.25 per page

2 37 CHI-KWONG LI and We have lnt(pa(x);n) = {pa(x)j = W(Cn(XI - A)). (2) Int(pA(A); fc) - W(Ck(XI - A)) = Pk(XI - A) for fc = 1, n - 1, n. Related to W(Ck(B)) is the concept of the kth decomposable numerical range of the matrix B defined by (see [5]) where /\ W (B) = ixck(b)x* : x G /\ hcn is decomposable and xx* = l}, Cn is the fcth Grassman space over C. Evidently, (3) W (B)CW(Ck(B)). It is known (see [4]) that equality in (3) holds for fc = 1, n 1, n. So (2) may be rewritten as (4) lnt(pa(x); fc) = W (XI - A) = W(Ck(XI - A)) = Pk(XI - A) for fc = 1, n 1, n. For general fc we have the following theorem. THEOREM l. Let A G C x be hermitian. for 1 < fc < n, Int(pA(A); fc) = W (XI - A) g W(Ck(XI - A)) = Pk(XI - A). All four sets are equal if and only if the set is convex. lnt(pa(x);k) = W (XI-A) PROOF. For B G CnXn, let B[k] denote the fc x fc leading principal submatrix of B. (see [5]), W {\I - A) = {det((xi - UAU*)[k\) : U unitary}. It follows that q(x) G W (XI A) if and only ifq(x) is the characteristic polynomial of a matrix of the form (UAU*)[k], where U is a unitary matrix. By a result of Fan and Pall [2], Int(pA(A);fc) = ^fca(ai-a). The inclusion W (XI-A)CW(Ck(XI-A)) is clear. Now consider W(Ck(XI A)). Since W(Ck(XI - A)) = W(Ck(U(XI - A)U*)) for any unitary matrix U, we may assume that Ai- A = diag(a - Ai,...,A - A ). It follows that Ck(XI A) is a diagonal matrix with diagonal entries EL-i (^ A»y ) with 1 < i\ < < ik < n. One easily checks that the numerical range of a diagonal matrix equals the convex hull of the diagonal entries. So W(Ck(XI-A)) = Pk(XI-A).

3 POLYNOMIALS AND NUMERICAL RANGES 371 Finally, if the four sets are equal, then clearly the set Int(pyt(A); fc) = Wk(XI A) is convex. Conversely, suppose Int(p>i(A); fc) = W^(Ai A) is convex. By the fact that A ) G lnt(pa(x);k) for 1 < i < < ik < n, we have Pfe(Ai-A)ÇInt(pA(A);fc), and hence the four sets are equal. D We show that the four sets are not equal in general by the following example. EXAMPLE 1. Let A = diag(l,2,3,4). p(a) = (A - 1)(A - 2), q(x) = (X - 3)(A - 4) G Int(pA(A); 2). However, the polynomial (p(x)+q(x))/2 has no real roots and thus does not belong toint(p (A);2). So Int(pA(A);2)#P2(A/-A). As mentioned in the proof of Theorem 1, W^(Ai A) is the set of all characteristic polynomials of matrices in the form (UAU*)[k], where U is unitary. For a hermitian matrix A, Wk(XI - A) is simply lnt(pa(x); fc). For general matrices A, we do not have such a nice equivalence form for Wk(XI A). Anyway, one may still consider the sets W(C;t(Ai A)) and Pk(XI A). In particular, we may extend Theorem 1 to THEOREM 2. Let A G Cnx be a normal matrix. W (XI - A) C W(Ck(XI - A)) = Pk(XI - A). The three sets are equal if and only if the set W (XI A) is convex. PROOF. Similar to that of Theorem 1. D For nonnormal matrices A, we have no inclusions between the sets H^C^Ai A)) and Pk(XI - A). Moreover, the set H^C^Ai A)) may fail to be convex as shown in the following examples. EXAMPLE 2. Let " 2" A = Example 3. Let Px(Ai - A) = {A} C {A - p: p < 1} = W(d(XI - A)). A = Ai-A ra J 1-1 A 1 A

4 372 CHI-KWONG LI and C3(Ai-A) = A3i + A A 11 1 L + 1 LJ Observe that p(a) = A3 + ax2 + A G W(C3(XI - A)) if and only if there exist z = (zi, X2,x3, X4) G C4 such that xx* = 1; Z1Z4 = ; X1Z3 + Z1Z4 + Z2Z4 = 2! xiz2 + Z1Z3 + Z2Z3 + Z3Z4 = a. By the second equation, zi = or Z4 =. Substituting equations, we have either (i) Z1Z3 = \ and Z2 = Z4 = ; or (ii) X2Z4 = \ and zi = Z3 =. It follows from the fourth equation that either a = \ or a =. So Pl(A) = A3 + \X2 + A, back to the first and third p2(a) = A3 + A G W(C3(XI - A)) whereas (pi(a) +p2(a))/2 W(C3(XI - A)). Hence W(C3(XI - A)) is not convex and cannot be equal to P3(XI A). Concluding remarks. 1. Example 3 gives a negative answer to the question about the convexity of the set W(adj(Ai A)) raised by Johnson [3]. 2. Observe that Ck(XI A) is actually a matrix polynomial. Instead of looking at W^Cft^Ai A)), one may consider the numerical range of a matrix polynomial Clearly, the set p(a) = XmA + + AAm_! + Am. W(p(A)) = {x(p(a))x* : x G C" and xx* = 1} is essentially the joint range of the matrices An,., Am, defined by W(Aq,...,Am) = {(xax*,...,xamx*): x G C" and xx* = 1}. As indicated in [1], the set W(Aq,, Am) may fail to be convex even for hermitian matrices An,..., Am, if m > Note that Ck(A) is a particular type of induced matrix (see [4]). Our results may be generalized to other classes of induced matrices. ACKNOWLEDGMENT. Thanks are due to Nam-Kiu Tsing for his helpful discussion.

5 POLYNOMIALS AND NUMERICAL RANGES 373 REFERENCES 1. Y. H. Au-Yeung and N. K. Tsing, An extension of the Hausdorff-Toeplitz theorem on the numerical range, Proc. Amer. Math. Soc. 89 (1983), K. Fan and G. Pall, Imbedding conditions for hermitian and normal matrices, Cañad. J. Math. 9 (1957), C. R. Johnson, Interlacing polynomial, Proc. Amer. Math. Soc. 1 (1987), M. Marcus, Finite dimensional multilinear algebra. I and II, Marcel Dekker, 1973 and M. Marcus and I. Filippenko, Linear operators preserving the decomposable numerical range, Linear and Multilinear Algebra 7 (1979), Department of Mathematics, University of Wisconsin, Madison, Wisconsin 5376 Current address: Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23185