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.
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