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Topological Aspects of Sylvester's Theorem on the Inertia of Hermitian Matrices
Author(s): Hans Schneider
Source: The American Mathematical Monthly, Vol. 73, No. 8 (Oct., 1966), pp. 817-821
Published by: Mathematical Association of America
Stable URL: http://www.jstor.org/stable/2314173
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TOPOLOGICAL ASPECTS OF SYLVESTER'S THEOREM
ON THE INERTIA OF HERMITIAN MATRICES
HANS SCHNEIDER,
Universityof Wisconsin, Madison
1. A set of nXn matrices with complex elements has a natural topology
associated with it. One may thereforelook for a topological interpretationof
some resultsin the theoryof matrices.We shall show that Sylvester's classical
theorem on the inertia (signature) of Hermitian matrices concerns the connected componentsof the space of all Hermitian matricesof fixedrank r.
Most of the argumentsused in the proofof our theoremare elementaryand
familiar.Yet our result does not appear in the literature.The reason may well
be that matrixtheoriststend to use "continuityproperties"as they arise, without formalizingthem, while topologistsdo not usually study equivalence relations on matrices. This note is offeredas an illustrationthat even on a fairly
elementarylevel, somethingis gained by looking forinter-connections
between
differentmathematicalfields.
2. Let n be a positive integerand let w= (ir,v, 5) be an orderedtripleof nonnegative integers with r+v+ a=n. Let E, be the diagonal matrix E,
, 1, -1, ...
=diag (1,
I-,-1, 0, * , 0) with r diagonal elements 1, v diagonal elements -1 and 8 elements 0. Sylvester's theorem ([7] p. 100, [3]
p. 83) on the inertiaof Hermitianmatricesasserts that foreach nXn Hermitian
matrix H there exists just one matrix E,, for which there exists a nonsingular
matrix X such that X*HX=E<,.
But can we pick out the triplew that occurs in Sylvester's theorem,without
use of that theorem and directlyin terms of the matrixH? One possibility,
whichwe mentionmerelybecause of its intrinsicinterest,is to proceed geometrically. Let V be the space of all positive n-tuples and associate with H the
quadratic form A: (x, x) =x*Hx. If it should happen that for some yE V,
(y, y) > 0 then also (x, x) > 0 forx = cy if aoz 0. Thus we have founda subspace
W of V of which (x, x) > 0 forall x#0. In otherwords,A is positive definiteon
W. Now suppose that 7ris the dimensionof a subspace W of largest dimension
on whichA is positiveand similarlysuppose v is the dimensionof a subspace W'
of largestdimensionon whichA is negativedefinite(i.e., (x, x) <0, if0 f xE W').
If 8=n-7r-v, then it may be proved that co=(wx,v, 5) is the w of Sylvester's
theorem,(see [1] pp. 148-150).
3. We shall use an entirelydifferentapproach. Since H is Hermitian all its
eigenvalues are real. We shall definewx,
v, 8 in termsof the eigenvalues of H.
DEFINITION 1. Let 7r(H)= r be thenumberofpositiveeigenvaluesofH, v'(H) = v
thenumberofnegativeeigenvaluesofH, and 6(H) = 8,thenumberofzeroeigenvalues
of H. Then theorderedtriplew=(r, v, 8) will be called theinertia of H. We shall
writew= In H.
817
818
TOPOLOGICAL ASPECTS OF SYLVESTER S THEOREM
[October
DEFINITION
2. Two Hermitian matricesH, K are inertiallyequivalentif
In H= In K. We shall writeH, {K.
The next definitionis standard.
DEFINITION 3. (e.g. [6] p. 99, p. 84). Two HermitianmatricesH and K are
conjunctive(conjunctivelyequivalent)if thereexists a nonsingularX such that
X*HX = K. We shall writeH, K.
It is evident that
matrices.
i
and
.
are equivalence relationson any set of Hermitian
kind,since theymay
4. Our next two equivalence relationsare of a different
be definedon any topological space.
If E is a topological space, the space is called connected if the empty set
and E are the only subsets of E whichare both open and closed (see [5] p. 117).
A subset U of E is connectedifand only ifit is a connectedspace in the topology
induced on U by E. Thus U is connectedif and only if,forany set FCE which
is both open and closed, either UC F or UCE\F, the complementof F in E.
This motivates
in E
DEFINITION 4. Let E be a topologicalspace. We call x, yCE connectable
ifforeveryopenand closedset U in E bothx, yE U or bothx, y EEU. We shall write
U
Xr..jY.
An arc in a topological space E is a continuous image of the interval (0, 1)
on the real line in the space E ([5] p. 139).
DEFINITION
5. Let E be a topologicalspace. We call x, yEE arc connectable
in E if thereexistsan arc in E joining x, y,i.e., if thereexistsa continuousfunction
f fromtheunit interval(0, 1) on thereal line intoE withf(O) =x and f(l) =y. WZe
shall writex,ay.
The followinglemma is a restatementof a well-knownresult (see [5] p.
141).
1. If E is a topologicalspace, thenxay implies thatxZy.
Proof. Let U be any open and closed set containingx and let f be a continuous functionof (0, 1) into E withf(O) =x and f(1) =y. Thenf-'(U) is an open
and closed subset of (0, 1) whichcontains 0, and the only such set is (0, 1) itself.
Thus y =f(1) C U and so x y.
LEMMA
5. Let S be any set of nXn matrices.We can norm S in many ways. For
example we can put A|I|=maxz,j IaijI for A ES. To turn S into a topological
space we choose as the open sets arbitraryunions of finiteintersectionsof all
cubes N(A, e)-={BCS: maxi,j Jb1j-aijI =IB-AJj <e}, with A ES and e>0.
Thus a subset T of S is open if and only if forA E T we can findon e = 0 such
that N(A, e) C T. Observe that S need not be a linear space, nor will our topological space S necessarilybe complete.
1966]
TOPOLOGICAL ASPECTS OF SYLVESTER S THEOREM
819
In this section, we shall consider the space N of all nonsingularcomplex
matricesnormedas above.
LEMMA
2. For all A, BEEN, A is arc connectableto B in N.
Proof. It is enough to prove that for all A EN, A 21, the identity matrix.
Choose a-so that etOAhas no negative eigenvalue, and set f(t) = eitA, 0 <t< 1.
If A belongs to N, so does ei,tA and clearlyf is continuous. Thus A aC. Next
set g(t) = (1 -t)C+tI, 0_ t ?1. Evidently g is again continuous,and since the
eigenvaluesof g(t) are ofthe form-y(t)= (1 -t)-y+t where-yis an eigenvalue of C
and herey is not negative,it followsthat y(t) 50 and so g(t) is nonsingular.Thus
C,aI. Hence A2a11,and this completes the proof.
6. We now require a lemma of a differenttype. Usually it is expressed by
assertingthattheeigenvaluesofa matrixare continuousfunctionsoftheelements
of the matrix.We shall state the result precisely:
, a. of multiLEMMA 3. Let A be a matrixwithdistincteigenvaluesa,
Let e> 0 and letF(ai, E) be thecirclewithcenter
plicitiesml,
, m. respectively.
ai and radius e. Then thereis a positivea-,such thateverymatrixB, for which
B -All <J-,has exactlymi eigenvaluesin thecircleF(ai, e).
*
*
,
Proof.Letp(t) =Xn+pn_1Xn-+ * * *+po, andq(t) =Xn-l+qn_,nXl+
.+qo.
We use a theorem on the zeros of a polynomial (see [4], p. 3). If the zeros
of p(X) are ai with multiplicitymi, i= 1, . * * i s, ai0aj, if i%j, and if (qj-pj)
n,X-1, where qj is sufficiently
small, then mi zeros of q(t) lie
<, j-O, 1,
in the circleF(aP, e). Now the eigenvaluesof A and B are simplythe zeros of the
are sums of products
characteristicpolynomialsdet (XI-A) whose coefficients
of elementsof A, and similarlyforB. Since addition and multiplicationofcomsmall o >0, lB
plex numbersis continuous,we deduce that forsufficiently
.
<O, implies that Iqi-pi I <,j=0,
, n-1, and the resultfollows.
Our proofof Lemma 3 is not really much of a proof,since it refersthe result
forthe spectra of a matricesback to the correspondingtheoremforthe zeros of
polynomials ("continuityof zeros of polynomials"). This latter result is deeper
than any othertheoremwe have used in this note, and we shall not attempt to
prove it here. We may note that in the application of Lemma 3, the matrices
A and B are both Hermitian.For normal,and thereforeforHermitianmatrices,
a more precise resultis given in [2]:
-All
LEMMA
019 .
* *
4. If A and B are normalmatriceswitheigenvalues ,,
a, , and
suitablenumberingof the eigenvalues
thenthereexistsa
A,n respectively,
such that
i
Iai-
l12<
;,j
I ai-
bij12.
This result rests on the famous theoremof Birkhoffthat the permutation
matricesare theverticesof the convexpolyhedron
ofdoublystochasticmatrices. For a proofof Birkhoff's
theoremsee [3] p. 97, or [2], and fora proofof
820
[October
TOPOLOGICALASPECTS OF SYLVESTER'S THEOREM
Lemma 4 see [2]. Of course, Lemma 4 implies Lemma 3 since
2
<n-21A -B 1
Es
I aij-bij
2
7. From now on our space willbe the space H? of all n X n Hermitianmatrices
offixedrankr. We shall firstexamine a trivialsituation.The space of all Hermitian l X1 matricesis just the real line R and hence HI is the real line with the
origin removed. The connectivitypropertiesof R and HI are quite different.
R is connectedand HI is not. Similarlythe connectivitypropertiesof H: will be
quite different
fromthese of the space of all n X n Hermitianmatrices.The reason forfocusingon Ht is that this space yields an interestingtheorem.
Notation. Let HEH,. Then the set of all KEEH' such that HLK will be
denoted by I(H), and the eigenvalue class I(H) will be called an inertial component of H,. Similarly we define C(H), U(H), A(H) to be the equivalence
a , respectively,and we call C(H) a conjunctive comclasses of H for c2u
ponent, U(H) a connected componentand A(H) an arc componentof H,.
THEOREM. Let H1' be the topologicalspace of all nXn Hermitian matricesof
Z coincide on H . Equivarank r. Then thefour equivalencerelations t a c a
lently,for any HCzHi, I(H) =C(H) =A(H) =U(H).
Proof. We shall prove that HLK implies H JK, H cK implies HaK,
HaK implies Hu K, and Hu K implies H,K.
(a) H{K implies H K: Suppose In H=In K=w=(ir, v, 6) say. It is
enough to prove that H,EW, where Es,,is defined in Section 1. (Note that
+v= r and that E,,WEH,.)Since H is Hermitian thereexists a unitary Y such
that Y*HY=diag (a,, * - *, a,,). By definition
of H, and since the oai are the
eigenvalues of H, it followsthat r of the ci are positive,v are negative and the
remaininga are zero. Replacing Y, ifnecessary,by the unitarymatrix YP, where
P is a permutationmatrix,we may suppose that Y*HY=diag (a,, - * *, a,)
where ai>O, i=1, * , 7r,ao+?<O, i-7r+l, . .. , r+v, and oa=O, i=7r+v
Thus if D=diag (Va/a,,. .. , \a,,, I-a,+,, . . ., a/-ar+, 1, *... , 1) and
X= YD-1, then X*HX=E,.
(b) H,ZK impliesH K: Let X*HX = K, whereX is nonsingular.Since by
Lemma 2 any two nonsingularmatrices are connected in the space N of nonsingularcomplex matrices,thereis a continuousfunctiont-+X(t) of (0, 1) into
N such that X(0) =I and X(1) =X. Further,transpositionand matrix multiplication are continuous operations. Hence the functionf(t) =X*(t)HX(t) is
continuous. But rank X*(t)HX(t) = r, since X(t) is nonsingular, whence
X(t)HX(t) EH'rl. Furtherf(O) = H and f(l) =K, and so H2aK.
(c) H, K implies H,K. This is just Lemma 1.
(d) HPK implies HLK. We shall prove the equivalent result that I(H)
contains U(H), forHCH,: Let KEI(H), and let In K-=c=(r, v, 8), and sup,Ir, of K are positive, the eigenvalues ao,
pose the eigenvalues ai, ij1,
1
.
19661
i=7r+1,
TOPOLOGICAL ASPECTS OF SYLVESTER'S
**,7r
THEOREM
+v are negative and the eigenvalues ai, i=r+v+1,
v
821
* *, n are
zero.
Let 0<e <min {Iai ai 0
By Lemma 3 there exists a neighborhood
N(K, u) of K in H' (thus each LCN(K, a) is, by assumption,Hermitian of
rank r) so that the spectrumof each LC2N(K, a) is contained in the union of
the n circles r(aci, e). It followsthat each L in this neighborhoodof K has at
least as many positive (negative) eigenvalues as K has positive (negative)
eigenvalues, i.e. 7r(L)_7r, and v(L) ?zv. But as LcEH', wr(L)+v(L)=r=r+v,
whence w(L) =iX and v(L) = v. Thus In L-In K. It followsfor each K EI (H),
there exists an N(K, a)CI(H), and so I(H) is open. Now II,\I(H) = U {I(M):
MEEH, MEEI(H) } and a union of open sets is open. Hence I(H) is also closed.
By the definitionof U(H), we see that U(H) is contained in every open and
closed set containingH, whence U(H) CI(H). This completes the proofof the
theorem.
}.
COROLLARY. The topologicalspace HR has preciselyr+ 1 distinctinertialcomponents(or conjunctivecomponents,or connectedcomponents,or arc components).
Proof. Obviously, each o= (r, v, 6) with 7r+v = r correspondsto one inertial
componentof H', and thereare just r+ 1 such w. By the theorem,each inertial
component is a conjunctive component (and a connected component and an
arc component).
8. It should be noted that in the proofof our theorem(a) is just a standard
proof that there exists an cosuch that Hc ,EW. The direct proof that this Xois
unique is simple (see [7] pp. 92, 100), but we do not need this proof. For distinct c, the correspondingE. obviously lie in distinctinertialcomponentsand
the uniqueness now followsfromthe equality of Z and on H,.
The concept of inertia may be extended to matricesthat are not Hermitian.
For some resultsin this directionsee [6].
,
I wishto thankMarvinMarcus,JudithMolinar,and RobertThompsonforcommenits
which
have helpedto improvethispaper.
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Algebra,Interscience,
New York,1957.
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3. M. Marcusand H. Minc,A surveyof matrixtheoryand matrixinequalities,
Allynand
Bacon, Boston,1964.
4. M. Marden,The geometryof the zeros of a polynomialin a complexvariable,Math.
SurveysIII, AmericanMath. Soc., 1949.
5. B. Mendelson,Introduction
to Topology,Allynand Bacon,Boston,1962.
6. A. Ostrowski
and H. Schneider,
Some theoremson the inertiaofmatrices,
J. Math. AiialysisAppl.,4(1962) 72-84.
7. S. Perlis,Theoryof matrices,Addison-Wesley,
Reading,Mass., 1952.