1.3. Eigenvalues and sums 45

1.3.3. Eigenvalue inequalities. Using the above minimax formulae, we

can now quickly prove a variety of eigenvalue inequalities. The basic idea is

to exploit the linearity relationship

(1.61)

v∗(A

+ B)v =

v∗Av

+

v∗Bv

for any unit vector v, and more generally,

(1.62) tr((A + B)

V

) = tr(A

V

) + tr(B

V

)

for any subspace V .

For instance, as mentioned before, the inequality (1.54) follows imme-

diately from (1.53) and (1.61). Similarly, for the Ky Fan inequality (1.56),

one observes from (1.62) and Proposition 1.3.4 that

tr((A + B)

W

) ≤ tr(A

W

) + λ1(B) + · · · + λk(B)

for any k-dimensional subspace W . Substituting this into Proposition 1.3.4

gives the claim. If one uses Exercise 1.3.3 instead of Proposition 1.3.4, one

obtains the more general Lidskii inequality

λi1 (A + B) + · · · + λik (A + B)

≤ λi1 (A) + · · · + λik (A) + λ1(B) + · · · + λk(B)

(1.63)

for any 1 ≤ i1 · · · ik ≤ n.

In a similar spirit, using the inequality

|v∗Bv|

≤ B

op

= max(|λ1(B)|, |λn(B)|)

for unit vectors v, combined with (1.61) and (1.57), we obtain the eigenvalue

stability inequality

(1.64) |λi(A + B) − λi(A)| ≤ B op,

thus the spectrum of A + B is close to that of A if B is small in operator

norm. In particular, we see that the map A → λi(A) is Lipschitz continuous

on the space of Hermitian matrices, for fixed 1 ≤ i ≤ n.

More generally, suppose one wants to establish the Weyl inequality

(1.55). From (1.57) that it suﬃces to show that every i + j − 1-dimensional

subspace V contains a unit vector v such that

v∗(A

+ B)v ≤ λi(A) + λj(B).

But from (1.57), one can find a subspace U of codimension i − 1 such that

v∗Av

≤ λi(A) for all unit vectors v in U, and a subspace W of codimension

j − 1 such that

v∗Bv

≤ λj(B) for all unit vectors v in W . The intersection

U ∩W has codimension at most i+j −2 and so has a non-trivial intersection

with V ; and the claim follows.