SIAM J. MATRIX ANAL. APPL. Vol. 9, No. 4, October 1988

(C) 1988 Society for Industrial and Applied Mathematics OO2

LINEAR PRESERVERS OF THE CLASS OF HERMITIAN MATRICES WITH BALANCED INERTIA* STEPHEN

PIERCE"

AND

LEIBA

RODMAN

Abstract. Let H(n) be the n2-dimensional real vector space of Hermitian matrices. Assume n is even and greater than or equal to 4. Let Tbe an invertible linear transformation on H(n) that maps the class of invertible, balanced inertia (signature zero) Hermitian matrices into itself. Then for some real number c 4 0, and an invertible matrix S, T(A) cS* AS or T(A) cS* A rS, for all A e H(n). T is also classified in the case where n=2.

Key words, linear preservers, inertia, Grassmannian

AMS(MOS) subject classifications. 15A04, 15A57

1. Introduction. Let H(n) be the set of all n n Hermitian matrices considered as a vector space of dimension n 2 over the real numbers R. For nonnegative integers r, s, such that r + s + n, we let (r, s, t) represent the inertia class of all A in H(n) such that A has r positive, s negative, and zero eigenvalues. We are particularly interested in the nonsingular balanced inertia class (k, k, 0) (so that n 2k is necessarily even). Suppose T is an invertible linear transformation mapping the class (k, k, 0) into itself. The purpose of this paper is to classify all such T. In previous related work [HR], [JP1], [JP2 ], the authors classified all invertible linear transformations T mapping the inertia class (r, s, t) into itself with the following exceptions: (i) the positive definite and negative definite inertia classes (n, 0, 0) and (0, n, 0); (ii) the nonsingular balanced inertia class (k, k, 0) with n 2k. In some cases, the assumption of invertibility of T could be removed without disturbing the result. We suspect this is always the case for nonsingular indefinite inertia classes when n >-_ 2. The problem is significantly more complicated for the definite inertia classes. For a discussion of results in this area see, for example, [C]. The following theorem is the main result of this paper. We assume n 2k. THEOREM 1.1. Let T be an invertible linear transformation on H( n ), where n >= 4, and assume that T( k, k, O) c k, k, 0). Then T is one of the following four types (here S stands for afixed invertible n n matrix)" (1) T(A) S’AS; (2) T(A) -S’AS;

.

(3) T(A)= S*ArS; (4) T(A)= -S*ArS. Note that Theorem 1.1 fails for n 2. Indeed, for a fixed a C, let T be the linear transformation on H(2) that multiplies a2 by a and a2 by It is easy to see that if a] > 1, then T is invertible and preserves the inertia class (1, 1, 0), but is not one of the four forms satisfying Theorem 1.1. As a byproduct we also obtain a characterization of the invertible linear transformations on H(n) that preserve the set of matrices with a k-dimensional isotropic subspace. THEOREM 1.2. Let T: H( n H( n be an invertible linear transformation, where n is even and greater than or equal to 4. Suppose that, for any A H(n) such that there is a k-dimensional subspace V C with x*Ay O for all x, y V, the matrix T(A) also enjoys this property (as before, k n/2 ). Then T is one of the four types described in Theorem 1.1.

-

Received by the editors July 30, 1987; accepted for publication (in revised form) February 5, 1988.

? Department of Mathematical Sciences, San Diego State University, San Diego, California 92182. The research of this author was partially supported by National Science Foundation grant DMS-0861959. Department of Mathematics, Arizona State University, Tempe, Arizona 85287. The research of this author was partially supported by National Science Foundation grant DMS-8501794. 461

462

S. PIERCE AND L. RODMAN

Indeed, the class of A H(n) such that x*Ay 0 for all x, y V, for some kdimensional subspace V c C n, coincides with the closure (k, k, 0) of (k, k, 0). It is easy to see (arguing by contradiction) that T( k, k, 0) c (k, k, 0) implies T(k, k, O) c (k, k, 0). Indeed, assume A a_ T(k, k, 0), where the invertible linear transformation T maps k, k, 0) into itself. Then A belongs to k, k, 0). On the other hand, (k, k, 0) is an open set and T is an open map. Thus A belongs to the open set T(k, k, 0), which is contained in (k, k, 0). As the interior of (k, k, 0) coincides with (k, k, 0), the matrix A actually belongs to (k, k, 0). Now use Theorem 1.1. The methods of this work are almost completely independent of [HR], [JP 1], and JP2 ], although we do rely on one particular lemma in [JP2 ]. We depend heavily on an examination of the Grassmannian G(n) consisting of the subspaces of C n. In the case where n 2, we have an entirely different problem. The conclusion of Theorem 1.1 is no longer true as noted above. To state Theorem 1.3, we first define for r, s e R, a linear map Dr, on H(2) by u- iv

b

b

ru- sir

As long as r[ and Is[ are greater than or equal to 1, Dr, preserves K(2), the set of indefinite 2 2 Hermitian matrices. If r s, we will just write Dr. Finally we note that the linear preservers of K(2) preserve the negative cone of the quadratic form ab u 2 v 2 obtained from the Hermitian matrix given above in the definition of Dr, s. This quadratic form has signature (1, 3, 0), precisely as in the form appearing in special relativity theory. THEOREM 1.3. Let T be an invertible linear map on H( 2 ), which maps K( 2 into itself. Then T is a product of maps of the type given in Theorem 1.1 and maps of the form Dr, with lr[, sl >-- 1.

.

2. The Grassmannian. in this section, we note some properties of the Grassmannian

G(n) consisting of all subspaces of C n. We endow C n with the standard inner product

(,) and the corresponding norm xll

xl = /

/

For V and W in G(n), we introduce the gap

Xn

o(v, w) P- Pwll, where Pv is the orthogonal projector on V and 11.11 is the operator norm, i.e., 11A is the maximum singular value of A. Now the gap 0 satisfies all the properties of a metric, thus turning G(n) into a metric space. PROPOSITION 2.1. With the metric O, G(n) is a compact (and hence complete) metric space. Moreover, G(n) has precisely n + connected components, each connected component being the set Gk(n) of all subspaces in C n of a fixed dimension k, O<=k<=n. PROPOSITION 2.2. Let Vm G( n), m 1, 2, and assume that

...,

limmoO(V, Vm)=0

for some V G(n). Then V consists ofprecisely those vectors x C n for which there is a sequence { Xm }, m 1, 2, where Xm C" and limm- Xm X and Xm Vm for m

1,2,-". These two propositions are well known and appear in, e.g., [GLR

or [GLR2].

3. Inertia classes. Recall that we are interested in the balanced inertia class 2k, k > 1. First we note that the closure of (k, k, 0), (k, k, 0) is the union of all inertia classes with positive and negative inertia not exceeding k. Alternatively,

(k, k, 0), with n

PRESERVERS OF BALANCED INERTIA

463

A e (k, k, 0) if and only if there is a k-dimensional A-isotropic subspace, i.e., a subspace V c C n such that (Ax, y 0 for all x, y e V. PROPOSITION 3.1. Let A, B k, k, 0), with A invertible. Then A and B have a common isotropic k-dimensional subspace if and only if (3.1)

XA+#Be(k,k,O)

,

for all # R. Proof. The result is obvious if A

and B have a common k-dimensional isotropic subspace. Thus suppose (3.1) holds. Without loss of generality, we assume that the pair A, B is in the canonical pair form with respect to simultaneous congruence (see [HJ] or [T]). Briefly, denote by Jp(o/) the p p Jordan block with eigenvalue O/. Then A and B are simultaneously direct sums of equal-sized blocks ofthe form +-Pr and PrK, respectively, where Pr is the r r permutation matrix satisfying Pu if + j r + 1; Pu 0 otherwise, and K is either Jr(a) for some real O/or Jr/2(a) (R) Jr/2(&) for some nonreal O/. It is easy to see that, in the cases when K Jr(o with real O/and even r or K (R) nonreal with the and matrices O/, Jr/2 Jr/z(g) +-Pr PrKhave a common r/2-dimensional isotropic subspace. Thus we need only consider the cases when K Jr(o with real O/ and odd r. As A e (k, k, 0), it follows that the number of blocks Pr in A with odd r is equal to the number of blocks Pr with odd r. This observation allows us to reduce the consideration to the case when

(:}.2)

r, (R)

(R)’r,(R) (--’,)(R)

(R) (--.P,,),

where rl, "", rq, $1, O/q,/1, /q are real (without loss Sq are odd and O/1, <- O/q; {31 >= >- q). As we can easily see, condition of generality assume O/1 <= (3.1) for A and B given by (3.2) is equivalent to the fact that the number of positive numbers in the list { k + O/1, -k +/3q} is precisely q for +/31, + every real k different from the -o/j s and/3 s. This can happen only if O/ -/3j for j q. (Indeed, letting k -o/ + e with small e > 0 and k =/31 e with small e > 1, 0, we conclude that O/1 +/31 0. Now apply induction on q.) Now the existence of a common isotropic k-dimensional subspace for A and B is evident (assuming q for the simplicity of notation. Letting r rl, s sl, we see that one such subspace is spanned by the vectors el, "", e(r-1)/2, er+l, er+(s-l)/2, e(r+l)/2 + er+(s+l)/2, where ej is [] the jth standard vector). For related results, see JR ], RU ], or RR ]. Now let T: H(n) H(n) be an invertible linear transformation such that T maps (k, k, 0) into (k, k, 0). Then clearly T(k,k,O) (k,k,O). PROPOSITION 3.2. Let A k, k, 0), B k, k, 0), and assume that A and B have a common isotropic k-dimensional subspace. Then the same is true of T(A and T(B). Proof By Proposition 3.1, hA + #Be(k,k,O) for all real X, #. Thus, X T(A + # T(B) T( )tA + #B) (k, k, 0), and the proposition follows from Proposition 3.1. For a given A (k,k, 0), let Z (A) c Gk(n) be the set of all k-dimensional Aisotropic subspaces. Proposition 2.2 shows that Z (A) is closed in Gk(n). For a given A (k, k, 0), let

,,k O/q,,-k

-

D(A) { B6(k,k,O) IZ(A)nZ(B)4 }. PROPOSITION 3.3. If T is as above, then T(D(A)) c D(T(A)).

464

S. PIERCE AND L. RODMAN

Proof Assume first that A e (k, k, 0). Let B D(A). By Proposition 3.2, T(B) D(T(A)), and we are done. Next, suppose that A (k,k,O) is singular. Let B D(A) and let V be a common isotropic k-dimensional subspace for A and B. Find a sequence Am (k, k, O) such that V Z (Am) for all m and Am "- A as m For example, we may assume that A=

0

AI

’

A2

A

-.

Am

A "m A 2m

where Alm, A2m are k k with invertible A lm and limm- Aim Ai, 1, 2. Obviously, B D(Am) for all m. By the part of the proposition already proved, T(B) D(T(Am)) for all m. Thus there is a common k-dimensional isotropic subspace Vm for T(B) and T(Am). By Proposition 2.1, choose a converging subsequence Vmp-- V for some V. As T(Am)" T(A), Proposition 2.2 shows that V is isotropic for T(B) and T(A). Since dim Vmp k, we also have dim V= k. Hence

T(B) D(T(A)).

1--1

4. More about the Grassmannian. With 2k n, consider the set Gk(n) of all kdimensional subspaces of C n, topologized by the gap metric. We introduce the standard structure of a real analytic manifold on Gk(n). n }. Let V, be a subset < a from { 1, Let a be a selection of k indices a < (chart) of G(n) defined as follows: V V, if and only if V is the column space of some n k complex matrix Xv whose k rows with indices a form the k k identity matrix. Note that Xv is uniquely defined by V. The set V, is open and dense in G(n). Define the map

f,: V,- R 2k2 by the property that f,(V) is the k k matrix formed by the rows of Xv other than the rows a. Clearly f is bijective and maps V homeomorphically onto R 2k2. Observe that, for any two selections a and/3, the setf( V, N Ve) is open in R 2k; in fact, the complement off (V, C) Va) is a real algebraic set. Also observe that the map

(4.1)

fof" f(V Vo)--R k

is real analytic in the 2k 2 real variables representing a point in R 2k. Note that Gk(n) is the union of V for all selections a. A set S Gk(n) will be called an analytic set if the following holds: For every V 6 Gk(n) and a chart V such that V 6 V, there exist an open neighborhood W off,(V) and a real analytic function g: W-- R such that X_ f-1(W)f"lS

if and only if g(f(X)) 0. Since (4.1) is real analytic this definition does not depend on the choice of the chart V,. Clearly, an analytic set is closed (in the gap metric). Finite intersections and unions of analytic sets are again analytic. PROPOSITION 4.1. Let A k, k, 0). Then Z A is an analytic set. Proof As Z (A) is a closed set in the gap metric, we must check the property that appears in the definition of an analytic set only for V e Z (A). Let V, be a chart such that V e V, and for notational simplicity suppose that a (1, k). Then V is the range (i.e., the column space) of

...,

PRESERVERS OF BALANCED INERTIA

for some k

465

k matrix X0. Write

A= then the range of

[A,lA2 A2] A22

I X] r Z (A) if and only if the Hermitian form [I

(4.2)

X*]A[x/]=0.

Write the (p, q) entry of X as xpu + iypq, p, q (4.2) can be expressed in the form

gs(Xpq,ypq)=O,

k. Then the system of equations

1,

1,

S

,S0,

where gs(Xpq, ypq) is a polynomial whose coefficients are real polynomials in the entries of A. Now set

gs Xpq ypq

g Xpq ypq

2

s=l

to satisfy the definition of an analytic set. A continuous map T: Gk(n) Gk(n) will be called analytic if for any two charts V, Ve the map

fTf 2: f.(X) R is analytic, where X { V V, ITV e Ve }. (The continuity of T ensures that X, and hence f(X), are open sets; thus the analyticity of the map fTfS defined on f,(X) makes sense.) PROPOSITION 4.2. Let T be an n n invertible matrix. Then the map 7: Gk( n Gk(n) defined by (V) { Txlx V} is analytic.

-

-

Analogously (using the charts) we define the analyticity of a map T: U R p, where U is an open set in Gk(n), and of a map T: R Gk(n) (in all these cases continuity of T is a prerequisite for analyticity). A closed set S c Gg(n) is called a real analytic manifold if for every V e S and a chart V, such that V e V there is an open neighborhood //of U such that //c V, and real analytic functions gl, "’", g22 on f,(//) exist with the following properties:

det[Ogi(t)] Otj

2k2

4:0,

t=(t,

tzk)ef(),

ai, j

f( S)= { tef() g+ (t)

gzk:(t)=O },

where p is some integer. It easily follows that the number p does not depend on the choice of gl, gzg (subject to the properties mentioned above) and that p is constant for every V belonging to a fixed connected component of S. The maximum of the numbers p will be called the dimension of S. It is a standard fact (see, e.g., [GN] or [W]) that any analytic set S can be represented as the union of a finite number of real analytic manifolds. The maximal dimension of these analytic manifolds will be called the dimension of S. Let S Gk( n ). Define

I’ (S) { A H(n) there is a subspace V S which is A-isotropic

466

S. PIERCE AND L. RODMAN

The set F(S) can be alternatively described as follows. Consider H(n) Gk(n), which is a real analytic manifold, and the set U of all pairs (A, V) Mn Gk(n) such that V S and V is A-isotropic. It is easy to see that U is an analytic manifold; then I’ (S) is the projection of U to the first component. We can show that F(S) is a semianalytic set (see, e.g., [L]). That is, locally F(S) is given as the set of solutions of a system of inequalities fl (A) >_- 0,... fs(A) >-- 0, where the f(A) are real analytic functions of the entries of A Hn. As any semianalytic set is locally the union of a finite number of real analytic manifolds, we can define the dimension of F(S) as the maximal dimension of a real analytic manifold contained in F(S). THEOREM 4.3. If S1, $2 c Gk(n) are analytic sets and dim SI < dim $2, then dim I’(S1) < dim F(S2). Proof Let V S, V2 $2, where V2 is such that the dimension of $2 coincides with the dimension of $2 in a neighborhood of V2. It is sufficient to prove that the dimension of F (W2 fq $2) is bigger than the dimension of I’ (WIN S ), where W and W_ are small neighborhoods of VI and V2, respectively. We can assume also that $1 and $2 are analytic manifolds. This follows from the fact that any analytic set in Gk(n) is a finite union of analytic manifolds. Applying an invertible linear transformation on C" which maps V1 onto V2, we can assume that VI V2 V, and set W WE W. Without loss of generality, assume that V is the column space of [Ik A0 T, where A0 is k k invertible. Thus

S=

{ [’] range

X (t) depends analytically on a parameter in

X(t)

where U

{ [’]

$2= range

Y(s)

(22

R ql is an open neighborhood of zero

Y (s) depends analytically on a parameter s in U2, where U2 c R q2 is an open neighborhood of zero

-

}.

-

Also X(0) A0; Y(0) A0; q < q2 and there are anal.ytic functionsfl X(U) for all t e U, and fz Y (s)) s for all s e U, f2: Y (U2) U2 such that fl (X (t)) We that assume may U2.

UI {(Xl, Vz {(x,

,xql,

,o)T-Rq2l--e
and e > 0 is suitably small, so that UI U2. Since A0 is assumed to be invertible, we can assume that Y(s) is invertible for all s e U2, and X (t) is invertible for all U. Applying to S the invertible linear transformation (which is a real analytic function oft U)

[, 0

0

y(t)X(t_)

]

teU,

Y(t) for all t U. (This follows from the equality F(X(S)) {X*AXIA I’(S)}, where Xis invertible.) Let H(k) be the vector space of all k k Hermitian matrices, and let F(k) be the vector space (over R) of all pairs of matrices (Z, Z2), where Z1 is k k Hermitian, we can assume that actually X(t)

467

PRESERVERS OF BALANCED INERTIA

and Z2 is any complex k k matrix. For any Y(s), s e U2, let Y(s) be the linear transformation from F(k) to H(k) defined by

(s)(Z, Z2)=-Y(s)*Z Y(s)- Y(s)*Z-ZY(s) Observe that

I can be identified with the graph of the linear transformation Y(s)"

I’ range

(4.3)

{(Z, I?(s)Z)IZf(k)}.

Y(s)

In this identification,

A=[AA2 A12] A22

from the left-hand side of (4.3) is identified with side of (4.3). Define the transformation

{(A22, AI2), A }

on the fight-hand

P: U2 -; L(F(k),H(k)), where L(F(k), H(k)) stands for the set of all linear transformations from F(k) to H(k) as follows:

F(s)

(s),

s- U2.

-

Then obviously F is real analytic. According to (4.3), the theorem will be proved if we can show that j0 has a real analytic inverse/-1. (U3) U3 for some neighborhood U3 of zero in R q2. By the implicit function theorem, it is sufficient to verify that

0P

j=l-."

q2

s=0

are linearly independent (here x, shows that

OF

(Z,Z2)= o

OY(O)

Ox

Xq2 are the coordinates of s

ZY(O)+ Y(O)*Z

OY(O)

Ox

So, if some linear combination of

is zero, say

=0, then, denoting q2

U=

OY(O) a j=l

oj R,

R q-). Computation

+Z, + Z2

Ox

Ox

468

S. PIERCE AND L. RODMAN

we have

+ Z:) U= 0 for all Z and Z2. Choosing Z and Z2 so that ZY(O) + Z I, and choosing Z and Z that so we Z again obtain U O. As 0 Y(O) iI, + Y(O) / dxj are linearly Z2 U*(ZY(O)+ Z)+(Y(O)*Z

’

independent,

-

COROLLARY 4.4. Let T: H( n) H( n be a linear invertible transformation such that T( k, k, O c k, k, O ). Then for every A e k, k, O we have

>= dim Z (A).

dim Z T(A

Proof Suppose not; then dim Z (A) > dim Z (T(A)) for some A e (k, k, 0). By Theorem 4.3 we have dim D(A) > dim D(T(A)). But dim D(A) dim D(T(A)), since an invertible linear transformation preserves the dimension of semianalytic sets. Thus /--1 dim D(A) > dim D(T(A)) contradicts Proposition 3.3. 5. Deductions from Corollary 4.4. To make use of Corollary 4.4 we need to compute dim Z (A) for certain Hermitian matrices A. LEMMA 5.1. IfA is an n n rank Hermitian matrix, then the dimension of Z A is 2k: 2k, where, as usual, 2k n. Proof It is easily seen that V Z (A) if and only if dim V k and V c ker (A). Thus Z (A) can be identified with the set of all k-dimensional subspaces in a (2k 1)E3 dimensional complex space. The real dimension of this set is 2k(k 1) 2k 2 2k. Recall that we always count real dimension. THEOREM 5.2. Let k >= 2 and A (1, 1, n 2). Then dim Z (A) 2k: 2k + 1. Proof. Without loss of generality we may assume that A is diagonal with precisely one l, precisely one -l, and n 2 zeros on the main diagonal. We shall consider only the chart U,o, where c0 (1, k). Then Z(A) N U, can be identified with the solutions X of the following equation:

[I Y*l(-Al

()A2)[]

=0,

where all the matrices in (5.1) have been partitioned into k k blocks, A1 Ipl -Iql Or, A: Ip2 q) -Iq2 @ 0r_, Pj + qj + rj n, j 1, 2, and P2 + ql P + q2 1. Rewrite (5.1) in the form

(5.2)

X*AEX=A

and observe that the necessary condition for solvability of (5.2) is that P2 q2. Three cases can occur:

(1) P2 q2 1, p ql 0, (2) p2=p 1, q q2=0, (3) p2=p =0, q q2 1. Consider case (1). Equation (5.1) takes the form (5.3) Write X

X* diag (1,-1,0,

[x,a], a,/3

1,

,0)X=0.

..., k and set

-" [ xl x21

X12]

>-- P and q2 >-

PRESERVERS OF BALANCED INERTIA

Let XI

(x13, Xlk) and X2 (x23, multaneously the following equations:

469

X2k). Then (5.3) amounts to solving si-

* diag (1,- 1))= 0,

(5.4) (.5,

,( x]

(5.6)

X{X

-X2

--0,

X X2.

First we shall count the dimension of the set of solutions X to (5.4). Clearly (5.4) is equivalent to

x 12=

Ix212= Ix==

x2 12,

=,

.11XI2 .21X22 "-0. First assume that x14=0. Then x2 =eUx22, cz(l{, and Xl2--X21Xl2/Xll-" e-’x22. Thus e- a x22

]

is given by five real parameters XII X22 O/(Xll and x being arbitrary complex numbers). Observe that

(5.7)

ker

*

span

ei

A similar argument shows that when x22 q: 0, is given by five real parameters and ker X* has the form (5.7). Finally, if X q: 0, at least either Xll q: 0 or x22 q: 0. In view of (5.7), the solution [X X2] r of (5.5) (where is considered as given) is given by 2 (k 2) parameters in X1; then (5.8) X2= eiaXl. If (5.8) holds, then (5.6) is satisfied automatically. The total number of parameters to describe the solution X of (5.3) is

5+2(k-2)+2(k-2)k=2k2-2k+ (the term 2 (k 2)k in the left-hand side describes the parameters in xij, >= 3, -< j =< k, which are absolutely free). Consider case (2). (Case (3) is obtained from case (2) by replacing A with -A and will not be discussed. Then

X * diag 1,0,

(5.9)

..., 0)S

diag 1,0,

..., 0).

Write

x= Xll X12 ]

x xJ’

and X22 is (k 2

1)

(k

1). Then (5.9) is equivalent to Xll[ 1, X 2X12 0, .el iX12 0. Thus X2 0, and the solutions X of (5.9) are described by one real parameter of X and 2k(k 1) real parameters of X2 and X22. The total number of parameters is 2k(k- 1)+ 1= 2k(k- 1). where Xl is

470

S. PIERCE AND L. RODMAN

THEOREM 5.3. Let A (k 1, k 1, 2). Then dim Z (A) >_- k 2 + 1. Proof. Let J be the k k matrix Ik- (R) O. Without loss of generality, we may assume that A J ) -J. It is sufficient to prove that the set of solutions X of

[I X*]

(5.10)

0

has dimension k 2 d- 1. Write

Y:x21

X12] X22

1. It is easily seen that (5.10) holds if and where X is (k 1) (k 1) and Xzz is only if X is unitary and X2 0. Since the set of (k (k 1) unitary matrices is (k 1) 2-dimensional, the total dimension of the set of solutions of (5.10) is (k 1) 2 + [2] 2k k 2 q- (the 2k dimensions appear because X21 and X22 are arbitrary). THEOREM 5.4. IfA e (k, k, 0), then dim Z (A) k 2. Proof. We can assume that A is a diagonal matrix with k ’s and k ’s on the main diagonal. It suffices to consider the intersection of Z (A) with one chart in the Grassmannian, say with the chart U,, where c (l, k). In other words, we must compute the dimension of the set of solutions X of

(5.11)

[I

X*][ Jl0 j20][]

=0’

where A J1 ) J2, and Jl and J2 are k k diagonal matrices with ’s and -1 ’s on the main diagonal. Equation 5.1 l) is equivalent to

(5.12)

X*J2 X= -J.

-J and J2 are congruent, and we may assume that J -J2. Now X satisfies (5.12) if and only if X is an element of the J2-unitary group. The (real) dimension of E] this group is k 2 (see, e.g., X.2 in [H]), and we are done. COROLLARY 5.5. Let T: H( n H( n be an invertible linear transformation such T( k, k, O) c k, k, 0). IrA k- l, k- l, 2), then T(A is singular. Proof Since the set of singular Hermitian matrices is closed, it is sufficient to assume that A e (k 1, k 1, 2). Now dim Z (A) >_- k 2 + by Theorem 5.3, and by Corollary 4.4 we have dim Z(T(A)) >_- k 2 + 1. In view of Theorem 5.4, the matrix T(A) must be Thus

-

if] singular, THEOREM 5.6. Let T: H( n) H( n) be an invertible linear that T( k, k, O) k, k, 0), where k >-_ 2. Then

(5.13)

transformation such

T(1, 1,n-2)(1, 1,n-2).

Proof Arguing by contradiction, we assume that T(A) is not in (1, l, n 2) for some A (1, l, n 2). We cannot have rank T(A)) because of Lemma 5.1 and Theorem 5.2. Thus we may assume (replacing A by -A if necessary) that T(A) has at least two positive eigenvalues. Since A belongs to (k l, k 2, 2), by Corollary 5.5 the matrix T(A) is singular. Using Lemma 3 in JP2 ], we choose a rank Hermitian matrix B such that i+(T(A+B))>i+(T(A)),

i_(T(A+B))>-i_(T(A)),

where i+ and i_, respectively, represent the number of positive and negative eigenvalues.

PRESERVERS OF BALANCED INERTIA

471

There are two possibilities: (i) n 4 (k 2 ). Clearly A + B e (2, 2, 0), but i+ T(A + B)) >= 3. This contradicts the assumption that T(2, 2, 0) c (2, 2, 0). (ii) n >- 6. Then A + B e (k- 1, k- 1,2). Again, from Corollary 5.5, it follows that T(A + B) is singular and i/ (T(A + B)) -> 3. Replacing A + B with A, we repeat our procedure, obtaining a contradiction if n 6 and iterating again if n >- 8. Eventually, of U] course, we will obtain a contradiction. It follows that T preserves the class (1, 1, k- 2). It then follows from the main result in JP2 that T has the required form, and hence Theorem 1.1 is established.

6. Proof of Theorem 1.3. Let Ei, j be the matrix with in position (i, j) and zero elsewhere. Suppose first that T maps every matrix in K(2) to a matrix in K(2), i.e., every matrix of rank is mapped to a matrix of negative determinant (such T do exist). Let S be the set of all A K(2) with Frobenius norm 1. Then S is compact and thus T(S) is a compact subset of K(2). It follows that there is an e, 0 < e < l, such that D,T still maps K(2) into K(2). If we assume e to be chosen as small as possible so that D,T maps K(2) into itself, then D,T will map some rank matrix in H(2) to a rank matrix. It is easy to see that D,T actually maps K(2) into itself, and that e > 0 (so that D,T is invertible). Replacing T by D,T, and applying congruence and negation (if necessary), we henceforth assume that T(Ell Ell. If the only rank matrices mapped to rank matrices by T are multiples of E, then an argument similar to that above indicates the existence of an e, 0 < e < 1, such that D,T preserves K(2), D,T(E) E, and D,T(A) has rank 1, for some A which has rank and is not a multiple of Ez. To within congruence, then, we may take T to fix both E, and E22. Now for any r e R, rE + (E,2 + E2 e K(2). It follows that the 1, 2) and (2, 1) entries in T(E2 + E2 are zero. The same is true, of course, for T(iEI2 iE21 ). If the only rank matrices mapped to rank matrices by T are multiples of E or E22, then we may argue once more that there is an e, 0 < e < 1, such that D,T preserves K(2) and that there is a rank matrix B such that DT(B) has rank 1, DT fixes both E and E2, and E,1, E22, and B are linearly independent. In addition, by performing a diagonal unitary congruence, we may also assume that B and D,T(B) are real, and that T fixes

EI2 + E21. Now suppose that

(6.1)

T( iE12 iE21

re_i

0

u + rye -i

Then for u, v e R,

T

u- iv

In order that T preserve K(2), it is necessary that

lu+rveil > [u+iv[ for any choice of u, v e R. This in turn, is equivalent to

(6.2)

0

r sin 0

v

0

472

S. PIERCE AND L. RODMAN

for all u, v e R. For (6.2) to be satisfied, the minimum singular value of the matrix in (6.2) must be 1. This occurs if and only if Irl >-- 1, and cos 0 0, that is, 0 +r/2. It then follows from (6.1) that T Dl,r. This establishes Theorem 1.3. REFERENCES

[C] [GN] [GLRI

[GLR2 H [HJ HR

[JP1 [JP2 [JR] L

M.-D. CHOI, Positive linear maps, Proc. Sympos. Pure Math., 38 (1982), pp. 583-590. B.C. GUNNING AND H. ROSSI, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, NJ, 1965. I. GOHBERG, P. LANCASTER, AND L. RODMAN, Matrix Polynomials, Academic Press, New York, 1982. Matrices and Indefinite Scalar Products, Birkhiuser, Boston, 1983. S. HELGASON, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. R. HORN AND C. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, 1985. J.W. HELTON AND L. RODMAN, Signature preserving maps on hermitian matrices, Linear and Multilinear Algebra, 17 (1985), pp. 29-37. C. JOHNSON AND S. PIERCE, Linear maps on hermitian matrices: the stabilizer of an inertia class, Canad. Math. Bull., 28 (1985), pp. 401-404. Linear maps on hermitian matrices: the stabilizer ofan inertia class, II, Linear and Multilinear Algebra, 19 (1986 ), pp. 21-31. C. JOHNSON AND L. RODMAN, Convex sets of hermitian matrices with constant inertia, SIAM J. Algebraic Discrete Methods, 6 (1985), pp. 351-359. S. LOJASIEWISZ, Sur les ensembles semi-analytiques, Actes, Congress. Internat. Math. Nice, 2 1976 ),

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pp. 237-241.

[RR] RU T

[W]

A.C.M. RAN AND L. RODMAN, Stability of invariant maximal semi-definite subspaces, I, Linear Algebra Appl., 62 (1984), pp. 51-86. A.C.M. RAN AND F. UHLIG, A note on a new description ofinvariant maximal nonnegative subspaces in an indefinite inner product space, Linear Algebra Appl., 71 (1985), pp. 273-274. R.C. THOM’SON, The characteristic polynomial ofa principal subpencil ofa hermitian matrix pencil Linear Algebra Appl., 14 (1976), pp. 135-177. H. WHITNEY, Elementary structure of real analytic varieties, Ann. of Math., 66 (1957), pp. 545-556.