1.3. Eigenvalues and sums 49

The equation (1.72) simplifies to ˙ u

∗ui

i

= 0, thus ˙ u

i

is orthogonal to ui.

Taking inner products of (1.71) with ui, we conclude the Hadamard first

variation formula

(1.73)

˙

λ

i

= ui

∗Aui.˙

This can already be used to give alternate proofs of various eigenvalue

identities. For instance, if we apply this to A(t) := A + tB, we see that

d

dt

λi(A + tB) = ui(A +

tB)∗Bui(A

+ tB)

whenever A + tB has simple spectrum. The right-hand side can be bounded

in magnitude by B op, and so we see that the map t → λi(A + tB) is

Lipschitz continuous, with Lipschitz constant B

op

whenever A + tB has

simple spectrum, which happens for generic A, B (and all t) by Exercise

1.3.10. By the fundamental theorem of calculus, we thus conclude (1.64).

Exercise 1.3.11. Use a similar argument to the one above to establish

(1.68) without using minimax formulae or Lidskii’s inequality.

Exercise 1.3.12. Use a similar argument to the one above to deduce Lid-

skii’s inequality (1.63) from Proposition 1.3.4 rather than Exercise 1.3.3.

One can also compute the second derivative of eigenvalues:

Exercise 1.3.13. Suppose that A = A(t) depends smoothly on t. By

differentiating (1.71) and (1.72), establish the Hadamard second variation

formula16

(1.74)

d2

dt2

λk = uk

∗

¨

Au

k

+ 2

j=k

|uj

∗Auk|2˙

λk − λj

whenever A has simple spectrum and 1 ≤ k ≤ n.

Remark 1.3.7. In the proof of the four moment theorem [TaVu2009b] on

the fine spacing of Wigner matrices, one also needs the variation formulae

for the third, fourth, and fifth derivatives of the eigenvalues (the first four

derivatives match up with the four moments mentioned in the theorem, and

the fifth derivative is needed to control error terms). Fortunately, one does

not need the precise formulae for these derivatives (which, as one can imag-

ine, are quite complicated), but only their general form, and in particular,

an upper bound for these derivatives in terms of more easily computable

quantities.

16If

one interprets the second derivative of the eigenvalues as being proportional to a “force”

on those eigenvalues (in analogy with Newton’s second law), (1.74) is asserting that each eigenvalue

λj “repels” the other eigenvalues λk by exerting a force that is inversely proportional to their

separation (and also proportional to the square of the matrix coeﬃcient of

˙

A in the eigenbasis).

See [Ta2009b, §1.5] for more discussion.