James V Burke.

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TECHNICAL REPORT

Differential Properties of Eigenvalues

James V. Burke
Michael L. Overton

Technical Report 579

September 1991

NEW YORK UNIVERSITY

r -

IT)
I

(x: >
w

M CD

Ph to

O

in

0)
â– H
4-1
i-'
G>

o

â–  a,

â€˘

c-i in
(0

O -

u C

-^ RU{Â±oo} by

w {z:d)= mf lim mf , (6)

76r(z," (6)

Akâ‚¬S(.!Â»)

where /it G 'Hci'^] for each At G ^{zÂ°)- The polynomials /i* can be taken to have the
form

/it(A, z) = (A - A*)" + Ckiiz){X - Ai)"-i + â€˘ â€˘â€˘ + Ci l,j = l,...,- 5, so ^j â€” for j = 3, ..., 1, we also have y^j â€” 0- I" the case f â€” 2, observe
that the condition Re yQr^ = is equivalent to the two conditions Re y^ri, < and
Im ylr^ = 0. Now, since â€” /?2 is the sum of the roots of Q{r), (16) follows. q

We apply this result to the evaluation of q''(zÂ°; d) where a is given by (1) with E(i)
and P defined in (5) and (8), respectively. First observe that we can replace the limit
infimum in (3) by limit since the perturbed roots of the polynomial (9) are given by
Puiseux series of the form (11) for some non-negative rational number p. Therefore,
this limit always exists and can only take the value +00 if it is not finite. Consequently,
o''(zÂ°; â€˘): C I-* R U {+00} and is given by

a'^{z;d)= inf Hm^Mlllz.^. (19)

We now state the main result of this section.

Theorem 2 Let P be given by (8) and choose d ÂŁ C". If any one of the conditions

is violated, then
otherwise

Re c'2(zÂ°)(i > 0, Im 4(2Â°)^ = 0,
cr(zÂ°)d = 0, j = Z,...,to

a''(zÂ°;d) = +oo;

Q''{zÂ°;d)> - Rec\izÂ°)d.
to

(20)
(21)

(22)

Moreover, if the rank of c'{zÂ°) is to, where c.C >-â–ş (C'" is given by

c{z) =

ci{z)

then equality holds in (22) whenever (20) and (21) are satisfied.
Proof Suppose +oo > 6 > a''(rÂ°; d). Then there is a 7 â‚¬ T{zÂ°,d) such that

a(7(c)) - a(.-Â°)

lim â–

0, Im c[2{zÂ°)d = 0, (30)

ci^.(zÂ°) 0. Consequently, (33) again follows from Theorem 1.

In order to establish equality in (33), it is again sufficient to exhibit the existence
of a curve -^ ^T(zÂ° ,d) such that

^.^ a(,(e))-a(z) ^ ^^^_Rec^ _ ^^ ^ ^^^^o^j (34)

We generate such a curve precisely as in the proof of Theorem 2 except now we must
choose the curve from V{zÂ°,d) so as to match the coefficients in (25) for each At â‚¬
A\{zÂ°,d). Just as before, it is the linear independence of the gradients {c'ki{zÂ°) : At ÂŁ
^i(zÂ°,d)} which guarantees that this can be done via the implicit function theorem.
Moreover, it is clear that we need only match the coefficients for At G A\{zÂ°,d) since
these are the dominant first order terms. D

3 Eigenvalues of Complex Matrices

Let W[(C"', C"^"] denote the set of mappings from â‚¬" io â‚¬"^" each of whose compo-
nents is an analytic map from (C" to ^k is said to be semisimple (nondefective).

Definition 4 Define the jth generalized trace of a square matrix A, denoted by

tr'j'yl,

as the sum of the elements on the diagonal of A which ts j â€”I positions below the main
diagonal. Thus one obtains the ordinary trace in the case j = I and the bottom left
element of the matrix in the case that j is the dimension of the matrix. If j exceeds
the dimension of A, take tr'-'M = 0.

10

Theorem 5 Let A e â– W[(C^ â‚¬"'*"] and choose 2Â° ÂŁ C . Suppose that A^"' = A{zÂ°)
has Jordan form as described above. Define

For each q = 1, . . .,t/ partition S~^ A\ S conformally with the partition of J and denote
its diagonal block corresponding to Jk by Bqk, k = l,...,ri, with each Bqk having
diagonal blocks Bqki corresponding to Jki, / = 1, â–  - â€˘ , "it- Then, for k = 1, . . .,t),

cUzÂ°) =

, forj = l,...,tk,

(36)

where the functions Ckj are as given in (7) for the factorization (6) of P(X,z) =
det[XI -A(z)] at z = zÂ°.

Proof Let max

{

a''{zÂ°;d) = +00;
ReE:=i'f,Eâ„˘\trn)B,â€ž

A* G Ao(zÂ°)

(37)

(38)

Moreover, if the vectors

for Xk e Ai(zÂ°,d) and j = 1, . . . ,tk,

(39)

are linearly independent, then equality holds in (38).

It should be observed that if any eigenvalue Ai G A\{zÂ°) is derogatory, then the
vectors in (39) cannot be linearly independent. In order to see this note that, for at
least one j between 1 and 0, Im XoC2(zÂ°)d = 0, (43)

c'j(zÂ°)d = 0, j = 3,...,{ ^^ tl^'^^'"^'^' -^' ^oc\{z')d\ , Â«/ (43) and (U) hold, ^^^^
P ( 1 ; _ 1^ ^^^ ^ otherwise.

Here it is understood that the function C2 is identically zero if to = 1.
Moreover, if the rank of c'{zÂ°) is \X(e)\-\Xol (47)

or equivalently,

26|Ao|e + o{â‚¬)>|A(Ol'-|AoP, (48)

is that

6|Ao|â‚¬ + o(â‚¬)>ReAo(A(f)-Ao). (49)

It follows from Lemma 1 that if either (16) or (17) with j/o = Aq, or equivalently, (43)
or (44), do not hold, then inequality (48) cannot hold for any (5 ÂŁ R. Since this is
independent of 7 6 r{z^,d), p''(zÂ°,d) = +00 if any one of (43) or (44) are violated
regardless of the value of ^(2").

Let us now suppose that (43) and (44) hold, i.e. (16) and (17) hold with j/o = Aq.
Then for every 6 > p''(zÂ°,d) there is a 7 â‚¬ r{zÂ°,d) such that

â– .n.'"*Â»-'"-Â°'

1

Online LibraryJames V BurkeDifferential properties of eigenvalues → online text (page 1 of 2)