Perturbation with two parameters

LINEAR

ALGEBRA

Perturbation

AND

ITS APPLICXTIOKS

with Two Parameters*

12. E. L. TURSER The

of

University

Madison,

Communicated

1.

Wisconsin

Wisconsin

by P. Y. Lax

If A and B are arbitrary

dimensional,

complex,

self-adjoint

inner-product

[l] states that all the eigenvalues functions [l]

on two parameters,

transformations

in a finite

V, then a theorem

of Rellich

and eigenvectors

of z, for z in some neighborhood

dependence in

space

of A + zB are analytic

of zero.

The question

of analytic

i.e., the case A + z,B + z,C, was raised

and the example

on z1 and z2 need not be analytic

was given to show that the dependence even if all transformations

involved

spectrum,

is analytic,

then

contour

everything

integral

The purpose

representation

are self-adjoint.

If A has simple

as can easily

for the projections

of this note is to show that

be seen from

the

on the eigenspaces.

this last assertion

has a partial

converse. Let

B(V)

denote

usual associative adjoint

the

product,

transformations

~2’ C S(V)

is analytic

of all automorphisms

and let S(V) in B(V).

denote

and eigenvectors

Institute

0

of self-

of elements

pair of transformations

numbers

Algebra

1969 by American

depending

by NASA

of the University

Linear

Copyright

the

B and

of .4 + z,B + z,C are represent-

* The work reported here was supported Chemistry

a collection

power series in zi and z2 in some polydisk

/z21< pZ, where pi and pZ are positive

Theoretical GP-5574.

of V with

the real subspace

We say that

for A if for every

C in &‘, the eigenvalues able as convergent

algebra

and

Grant XsG-275-62

of Wisconsin,

Its

Elsevier

IzrJ < pi,

upon the pair at the

and by NSF Grant

Applications

Publishing

2(1969),

Company,

l-

11 Inc.

2

K. E. L. TURNER

B, C.

We

suppose

that

moduli of numbers in S(V) From

are subspaces

theorem

Of course,

each have dimension

the collection

of all elements

Vi will be analytic follows that

for A.

can

analyticity.

with respect 2 modulo

be added

to any

forbiddingly

4 and above. complicated

under the assumption is guaranteed specified

by a Jordan

with no product 2.

Let

then

subspaces

we assume in what at most three and &

the span of A and the identity Factoring

analytic

However,

I,

out A and I is necessary

subspace

without

affecting

to the corresponding

our method

of the spectrum

of proof

For V having of A is obtained

or Lie multiplication In a final section

A be a self-adjoint

projections

property

result becomes

arbitrary

in Section 3

which, in turn,

in ,d, provided G? contains we return

transformation

multiplicities E,, E,, . . . , E,,

in

m,, m2, . . , mk respectively.

of A by ,ui, ,I_+ . . . , ,uu,, numbered

eigenvalues

under

to subspaces

&

defined.

A,, &> . . . , Ak having spectral

each is invariant

to A and &.

that & has a transitivity

transformations.

complement

if V is a direct sum

to each subspace,

in higher dimensions.

finite dimension the simplicity

one is

subspace

with the same reducing

We know of no counterexample

in dimensions

invariant

Further,

Vj, such that

of S(V)

to

consider

with dimension

2 that if V has dimension

with dimension

they

any &

a common

when restricted

then A must have simple spectrum. since

have

To avoid such degeneracy,

V is irreducible

It is shown in Section is analytic

We shall, therefore,

of A?’ to Vi and its orthogonal

spectrum

respect

of S(V).

1, & will be analytic.

Va 0 * * * @ V, of subspaces

Vi 0

A, and A has simple

with

It is clear that the linear span

one sees that

if A and &

V, and the restrictions

V,C

is absolute

set is again analytic.

sets which

Rellich’s

analytic.

convergence

and norms of vectors.

of an analytic

only analytic

the

V with and We

to include

eigenvalues

corresponding

also denote

the

multiplicity,

so

that ,ui = ,ua = * * . = ,u,,,~= il,, and so on. Supp ose that

~?HEOREM 1. for

If dim & = 2 mo&?o

A.

dim V < 3.

Let x2 C S(V)

be analytic

the span of A and I, then A has simple

spectrum. We

Proof. dependent

dim V = 3. Linear

shall

derive

of the dimension

Algebra

some

For lower dimensions and

Its

necessary

conditions

which

are

in-

of V. These results will then be applied when

Aj$&xztions

the proofs are trivial.

2( 19f39), l-

11

PERTURBATION

Let

8

WITH

TWO

and C belong

assumption,

A

P’(ZP 4 = orthonormal.

2

PhRAME’~EIIS

to &

has eigenvalues

3

and consider ;Iti’(zl, z2) =

A = c

A + z,B -t_ z,C.

W for i=l,Z,. 21p 22 p+ep,

. *, n. We may A+(i) = ;lci)4ci) and equating

Expanding

of like powers

z1Pz24 we obtain

$“‘)$‘“A 10 P 14

(C -

#& are eigenvectors

is an eigenvector

(A -

I$&;

= 0, 1%.

= I#&).If we set i = 1,

we have

(A -

I,)#;

= -

(B -

,I,) = 0, yields E,(B

-

P,P I

01

of A for the eigenvalues

Thus A$ is one of the &‘s, say ;1,, and hence E,&$

which, since E,(A

{f$@‘} 00 is coefficients

the

$“‘)$‘“‘~_

for $ 3 0, q > 0, and 1 < i ,( 1%. The first equation,

p = 1, and q = 0 in (2.1)

assume

the equations

(A _ jlM)$‘%’ = _ (B 00 PY

shows that the vectors

By

zipzsq)La~and eigenvectors

of E,BE,

in E,V.

IL(110))&2

-

&$E,r$~

Similarly,

(2.2) That

= 0.

for appropriate

is, &,’

numbering,

+$/ (i = 1 2 ml) are eigenvectors of E,BE,. By the same reasoning, + (1’ of E,CE, and thus E,BE, and E,CE, 00’. . .> &$“” are eigenvectors commute.

This observation

has also been made by theoretical

working with double perturbation We assume E,V,

now that

theory

and E,CE,

E,BE,

have simple

and let P,, P,, . . ., P,,,, be the associated

to act in V. Further,

we normalize

in

projections,

considered of I so that

For

superscripts

to ;1(l’ and #l’.

the

eigenvalues

A, B, and C with multiples

Ai = apd = A$ = 0. referring

chemists

(cf. [2, p. 2951).

remainder

of this Returning

section to (2.2)

we omit

the

we see that

where 2, = 23n_? Aj-lEj, or (I - E,)& = (1 - E,M,, = - G%o M,&,,,, denoting - Z,B by M,. Analogously, (I - El)& = - Z,C#J~,, = M,~,,.

From

Setting

$ = 2 and q = 0 in (2.1) gives

(2.3) we obtain

il,, = (&, 0% -

c:“2

Pr)$,o

and

B,M,&,) = -

Y,BM,400,

If we normalize

the eigenvectors

where

Y, =

satisfy

(d’“‘, &d) = 1, then for each 1 < i < ml, P&

($&lPt.

Li~tear Algebra

ad

4(i) to

= 0 for $ > 0 or

Its i4pplications

2(1969),

I-

11

4 q > 0.

R. E. L.

Denoting

-

TURNER

by N,, we can write

Y,B

%&I

(2.4)

= NP&lcl

and have (I-

E,)&,

= (Ml2 + JJ,N$f,)&.

(2.5)

Analogously, %#JO1=

N2M2400

(2.6)

+

(2.7)

and (1 -

4Mo2

=

CM22

M2N2M2Mxi~

with N, = -

Y,C =

2 (A$-‘P,C. i=2

From ‘%, we conclude

noting

= -

W,,

-

%”

+ Moo

(2.8)

that

that P,BN, (I-

-W,,

In addition,

=

from

(E, Continuing

= 0 for i = 1, 2, and can write (M,M,

+

(2.8) we obtain

PI) [B(M,

=

P”&Q2

+

+

M2M1 +

the necessary

+ N&z)

in this manner,

&$2o

MP2M2

+ CM,

M,Nd4,%m

condition

+ N,M,)l&,

= 0.

we find

N,M,N,M,

+

k4w

~M,+,,)YIN,M,)~,,

from Eq. (2.1) with p = 3, q = 0, and from 452,

=

by substituting Linear

Algebra

-

WI

-

from above,

and Its

Applications

c+,ll +

~2074,

+

obtain Z(1969).

I-

11

4,&l

+

~2,h~

(2.9)

PERTURBATION

-w,,

=

WITH

N,(M&f,

+ +

+

N,,(M12

+

b$,,~ ~M,ho)Y,~,M,)+,,

+

(Am

where IV,, =

PM2

M,N,M,

+

Y,C.

-

P.XRAMETERS

M,N,M,

+

if we interchange -

TWO

M&f, +

+

~,N,~1ha

N,M12 +

+

(hm

CMlMxJ(YlMl+

N,M,N,M, BM,&,)(Y,M,

YP2M2Mm

(2.10)

YlNlM2)hm

By symmetry

we must get the same vector

B and C, and the subscripts

1 and 2, letting

Thus (2.10) gives us a second necessary

Y,B.

+

condition

El& N,, =

relating

B and

C. If dim I/ = 3, the

two necessary

If A has just one eigenvalue in I/’ = E,V

and therefore

A has an eigenvalue

B in &‘, such that E,BE,

priate

in E,V.

choice

will yield

Theorem

1.

in JZZmust commute Suppose

A and &’ are reducible.

ii, with m, = 2 and that

be a C in &, independent spectrum

conditions

then any two elements

now that

there is a transformation

has simple spectrum

in E,V.

Then there will

of i? modulo A and I, such that C also has simple

Since E,BE,

and E,cE,

must commute,

of a basis in I/ and suitable

normalization

by an approwe can obtain

matrices

_

-

Cl3

523

such that A + z,B + z,C has analytic B and C will be linearly computation

eigenvectors

independent

shows that

0 and eigenvalues.

and irreducible

the necessary

condition

with A.

Also, A short

(2.9) can be put in the

form (‘23

Only one factor linearly E,&)

in (2.12)

dependent. = ~$2,

b23)(C13

-

‘13)

=

(2.12)

‘.

can be zero, for otherwise

Letting

q5g and 4% span E,V,

B and C would be one computes

that

where

a = ibaa(2C&a + -

-

h3j2C23c13

b;3c,3613

-

(b,3c,3

Linear

+

b236,3(lb,3j2

+

-

lb23i2)

+

C23ib23j2613

C13613)b23613~

Algebra

and

Its

Applications

2(1969),

l-11

R. E. L. TURKER

6 The second condition

on B and C is obtained by interchanging

the letters

b and c above to obtain G and setting (x = Z:. We may, without generality, C 13

=

The condition

b.

obtained

2/b2a/2~2a+ b&c,, -

from cc = Z then takes the form

~b2,~2b2,= 21~,,1~b,, + c&623 -

We assume now that b,, is real. vector vanish,

$$

by a suitable

factor

a contradiction.

then for some real t, C, = tB + (1 -

C, =

with c a real number.

Condition

If c23 = 6 + iv, with 6 # b,,,

0

b

0

1

ic

-ic

0i -

b& =

If 5 = b,,, since q # 0,

B, and C, of B and C with

(2.13) becomes

(e -

Li)3= 0 in this case,

a contradiction.

We must still examine having

simple spectrum

matrix

as above

the situation

in E,V.

when &

contains

no element

We may then assume that A has a

and that

the pair B, C being linearly independent

and irreducible with A. Multiply-

ing the basis vectors by factors, if necessary, real matrix,

of

c,J3 = 0 or

With C, in place of C, (2.13) becomes ib&

there are real linear combinations

again yielding

(b,, -

t)C has the form

ic3 or bi3 + c2 = 0, which is impossible.

where e is not real.

the irreducibility

become

0

( 6

the basis

The entry b,, cannot

one.

If c2a were real (2.13) would

b, = c23, again providing

1~~~1~~~. (2.13)

If it is not, we can multiply

of modulus

for (2.13) would give c2a = 0, contradicting

A, B, and C.

c2b2, -

loss of

assume that the second factor of (2.12) is zero, so that b,, =

we may assume that B is a

and by taking a real linear combination

that C has only two nonzero

entries.

Without

set cl3 = 0, which means b,, # 0. Returning Linear .dlgebra and Its Applicatiom

2(196Y), l-

of B and C, assume loss of generality,

we

to Eq. (2.1) we find as before 11

PERTURRhTIOS

WITH

TWO

PARAiMETERS

we obtain

and similarly

The

same

A short

holds

sequence.

and thus

shows that

contradiction.

eigenvalue

We see then

and the theorem

commute.

b,, = 0 is a con-

For

EiCEi commute

finite dimension

in E,V for i = 1, 2,

of eigenvectors

cf_r

E,BE,

with respect

B in &‘.

a double

are

elements

and

eigenvectors

of

of J&’ be represented

also

by

We say that &’ is tra&tive

bklbll12 # 0 for three

C = (cl?) in &’ with c,~,,~f

and let & be analytic

. , k. Thus I/ has an orthonormal

Let

to this basis.

satisfyring

have

B, C in ~2, E,BE,

&(‘1 (1 < i < n) which

for any

PROPOSITIOK.

A cannot

any pair of transformations

basis

B = (b,) in &

that

is proved.

Xow let I/ have arbitrary

for A E S(V).

matrices

and E,CM2E,

E,BM,E,

b,,b,, = 0, and thus

Now, using the pair B + C and C we find b,,c,, = 0, which

is the final

3.

for 4%

computation

integers

if for any

k, I, m, there

is a

0.

Sufipose

.nC is

analytic

for

A and transitive.

Then A

has simple spectrum. Divide

Proof. defining

the

eigenvalues

The relation

-

is obviously

reflexive

of ,G? implies that of the relation. subspace

of A

into

equivalence

classes

by

a relation ,uI “1~~ if there is a B = (bij) in ~9 such that b,, # 0.

E,V, two equivalent

be equal. represented

of the ,u; into

by a matrix

the diagonal has simple

eigenvalues

Now, if the eigenvectors

with the partitioning

A and sl

and symmetric,

Since any B E d

with nonzero

and the eigenvalues

are irreducible,

there

and the transitivity is “diagonal”

with different

subscripts

q5$ are renumbered equivalence entries

cannot

to correspond

classes,

each B E d

only in fixed

of A in each block

in each

“blocks”

are distinct.

can be only one block

is on

Since

and therefore

A

spectrum. I_.i~rear.4lgebra

and Ifs ilpplications

2(1969),

l-

11

8

R. E. L. TURSER

The space S(V) multiplications

can be made into an algebra

that

are

possible

are

and the Lie product

+(BC + CB)

THEOREM

2.

S(V)

Suppose

analytic for A, and that d

the

in many

Jordan

[B, C] = i(BC -

contains

ways.

product

Two

B * C =

CB).

a Jorda?z subalgebra which is

contains an element D with sim@le spectrum.

Then A has simple spectvtrm. Proof.

shall show that

We

~1 must

First we show that &’ must contain spectrum

and commuting

with A.

D, and contains

no proper invariant

all Wi have dimension spaces, spectrum

in ~2.

Then

for arbitrary

Let

9

contains

in n will

to W, will be A, and D,, respectively.

We

E different

That

in

means

by A, and D,

subalgebra

of B(W,)

generated

by A, and D,

Thus

2.

Thus

of the identity,

9” of 9 in B(W,)

it must contain an orthogonal 1r in W,.

This, however,

that A, and D, have no common invariant

2’ = (&r}

the symmetric

generated

If the commutator

from zero and the identity

it follows from a general algebra

and thus that ZZZ’ contains generated

the assumption

IV,.

all of S(W,), of S(W,)

more than multiples

subspace

one of the sub-

one, and let A, and D, be

and F, * D = F,DF,

multiplication.

would contradict

Suppose

If

subalgebra

and let 9 be the complex

projection

A and

T E S(V).

be the Jordan

under associative

than

under

to A and D.

F, of V on W, is a polynomial

shall show that A, and D, generate F,TF,

common

F, * A = F,AF,

be in & and their restrictions

V orthogonally

Since D = D + yI has simple nonzero

for some y, the projection

and therefore

subspace

greater

of A and D to IV,.

S having simple

each Wi is invariant

one, we can set D = S.

say W,, has dimension

the restrictions

a transformation

We can decompose

such that

as IV, @ W, @ * * * @ W,,

be transitive.

elements

theorem

and

since

(9”)” = 9,

in 9 constitute

9 = B(W,). However,

S(W,).

(see [2, p. 3071) that the Jordan

by A, and C, consists

of the symmetric

elements

subin

9’ = S(W,).

It follows from the argument one orthogonal

projections

which sum to F,.

Applying

Wi we see that & contains P,,i

= 1,2,.

Then

S =

..) n, satisfying

Linear

Algebra

crZ1

above

that

which commute

the same argument a family

,d contains

a set of rank

with each other

and A, and

to the remaining

of orthogonal,

subspaces

rank one projections

PIP3 = 6,,P,, P,A = AP,, and cFZ1

iP, E .d has simple

and Its Applications

spectrum

2(1969),

l-11

and SA = AS.

P, = I.

PERTURBATION

WITH

The eigenvectors for a suitable must

T\VO PARAMETERS

$00 (‘1 defined

numbering

be transitive,

b,, # 0 and cl,,, f

P&i

suppose

some

.d contains transformations

Let

It follows

eigenvalues

4.

(sl -

of

having

The transitivity

In this section

= bklclm+ 0. theorem.

taken

I times

be denoted by s,.

THEOREM 4.

where S E S(V)

sj2) are all distinct

has b,i # 0 for some pair i # i,

[S + ~0,

B],

of .F4 then follows analytic

of Theorem

Let d

(I = 1, 2,. . .) yields

easily.

for A if A +

cz=i

z,Bj has

for any three elements B,, B,, and B, 4 below

be analytic for d

commutes

are admittedly

awkward,

su$$ose

spectrtim,

of indices

that for each pair

are not simultaneously

reducible

and m = i # j,

when restricted

to

Then A has simple spectrum.

(E, + E,)V. We

use the notation

Let G be any element

and results of the previous

in JZZ, normalized

so that

real cc, C = G + CXSwill have simple spectrum

to E,V.

Letting B = .$ = S P,).%Y7,M,&-,

= 0.

in (E, -

For

when restricted

s,l, the necessary condition

Since S is invertible

sections.

(&,,, G&,,,) = 0.

suitable (E, -

space

and contain AZ, S, S2, . . . , S’”

and has simple

with d

Further

and (Ei + Ei)d

Proof.

for

of this paper.

maxizJ(mi + ml). A

by

Then for

but then, the theorem is what we are able to prove in a reasonable with the methods

with

only the (i, i) and (i, i) entries different

we call d

The hypotheses

where S commdes

of S be denoted

that if B = (b,) E&

analytic eigenvalues and eigenvectors in &‘.

S and 9,

si) + c((s,~ -

linear combination

a B E & with a matrix from zero.

the following

[C, . ., IC, [C, B]], . .]

Let the distinct

then a finite

To see that JX?

R = (bti) and C = cij having

Then A has simple spectrum.

real tc, the quantities

i # j.

of S and

be a Lie subalgebra of S(V) which is analytic jor

A and has simple spectrum. Proof.

for 1 < i < n.

contains

we can obtain

Let d

THEOREM 3.

[C, B],.

= $$

d

* (P, * C)) hasamatrixgijwithg,,,,

In a similar manner

Su@ose

must be eigenvectors

0 for some triple k, 1, ?n. Then it is easy to check that

G = (P, * (P, * B)) * (P,

A.

above

9

(2.9) becomes

P,)V,

it follows

that ~2*2bl=

Linear

-%#b,

Algebra

=

(4.1)

0.

and Its Applications

2(1969),

1- 11

10 In

R. E. L. TURNER

the

fact,

related

identity

(E, -

P,)GZ,G&

= 0

for all real tc.

E,(G + aS)Z,(G + ccS)E, = E,GZ,GE,

with A = A + z,S + z,G + .zaA2, we would

holds,

Had

since

we started

have found (4.2)

for zs near zero, from which it follows (E, -

J’,)GEjGA,o = 0

The second

for j = 2, 3, . . ., n.

that

condition,

~s~r~sAXl+

(4.3)

arising from

~,,~,~,2&l

If we again add zaA2 and multiply

(2.10), becomes (4.4)

= 0.

through

by ((E, -

we can

P,)CEJ2

obtain (E, for i -

P,)CE,CE&C&,

+ (E, -

2, 3, . . ., a, from condition

(4.4).

P,).%F,CE,CE,C+,

We still let G be any transforma-

tion in L-Z!and assume that S has no zero eigenvalue. S + cll for a suitable

u.

# 0 for j = 2, 3, . . ., m,,

The identity be reduced

Otherwise we use

If we fix i with 2 < i < n, then there exist

real numbers tlr, . . . , uZmsuch that G = G + 0, PjePj

(4.6)

= 0

cEY!r u,S” satisfies: P,GP,

and E$Ei

= DE, for some

=

real p.

(4.5) then holds for C = G and, with the use of (4.3), can to (E, -

P1)~E,GE,SE,G+o,,

= 0

or, equivalently, (E, Equation

(4.6) holds

combination choose

cr=r

with

P,)GE,SE,G&, S replaced

ykSk in place of S.

the yk so that Ei cy=, (E, -

= 0.

by Sk and therefore, Suppose E,P, = P,.

with

any

Then we can

ykSk = P,, yielding P,)GPjG+,,

= 0.

(4.7)

If G is represented by the matrix g, then for 1 = 2, 3, . gljgj, = g$rj

(4.6)

= 0 is a consequence.

Starting

, m,, the condition

with +(2), q%(3),. . . , q@)

we

find that gLjgkj= 0 whenever k < m,, 1 < ml, mi < j < mi+l, and k # 1. Thus each column Linear

Algebra

and

of the matrix Its Applications

representing

2(1969),

1- 11

E,GE,

has at most

one

PERTURBATION

nonzero

1VITH

element

TWO

in each column.

Rephrasing

and using symmetry

one sees that

at most one nonzero

entry

is a linear

11

P.4R.iMETERS

the arguments

the matrix

in each row and column.

space the position

all G in &‘. This last condition

of a nonzero

given above

for each block

entry

Moreover,

must

EiGEj has since ~2

be the same for

shows that if the multiplicity

of d, is greater

than one for some 1, then (E, + E,) V will be reducible for A and (EL + E&d for each

i # 1.

Thus

A must

have

simple

spectrum.

REFERESCES 1 F. Rellich,

Perturbation

University, 2 P. M

Cohn,

theory of eigenvalue problems,

Lecture notes, New York

1953. Universal Algebra.

Harper

and Row,

Sew

York,

1965.

3 J. 0. Hirshfelder, 1%‘. Byers Brown, and S. T. Epstein, Recent developments in perturbation theory, in Advances in Quantum Chemistry, Vol. 1, Academic Press, 9ew

York,

1964.

Received January

17, 796X

Linear Algebra and Its Applications

2(1969),

1- 11