Lemma 2.2 Given two n× n matrices G = {gk,j} and H = {hk,j}, then
|GH| ≤k G k ·|H|. (2.19)
Proof. Expanding terms yields
|GH|2 = n−1X
i
X
j
|X
k
gi,khk,j|2
= n−1X
i
X
j
X
k
X
m
gi,k¯gi,mhk,j¯hm,j
= n−1X
j
h∗jG∗Ghj,
(2.20)
where hj is the jth column of H. From (2.10) (h∗jG∗Ghj)/(h∗jhj)≤k G k2 and therefore
|GH|2 ≤ n−1 k G k2 X
j
h∗jhj =k G k2 ·|H|2.
2 Lemma 2.2 is the matrix equivalent of 7.3a of ([13], p. 103). Note that the lemma does not require that G or H be Hermitian.
2.3 Asymptotically Equivalent Matrices
We will be considering n×n matrices that approximate each other when n is large. As might be expected, we will use the weak norm of the difference of two matrices as a measure of the “distance” between them. Two sequences of n× n matrices An and Bn are said to be asymptotically equivalent if
1. Anand Bnare uniformly bounded in strong (and hence in weak) norm:
k Ank , k Bn k≤ M < ∞ (2.21) and
2. An− Bn= Dn goes to zero in weak norm as n→ ∞:
nlim→∞|An− Bn| = limn→∞|Dn| = 0.
Asymptotic equivalence of An and Bn will be abbreviated An ∼ Bn. If one of the two matrices is Toeplitz, then the other is said to be asymptotically Toeplitz. We can immediately prove several properties of asymptotic equiv-alence which are collected in the following theorem.
Theorem 2.1 1. If An∼ Bn, then
nlim→∞|An| = limn→∞|Bn|. (2.22) 2. If An ∼ Bn and Bn ∼ Cn, then An∼ Cn.
3. If An ∼ Bn and Cn∼ Dn, then AnCn∼ BnDn.
4. If An ∼ Bn and k A−1n k, k Bn−1 k≤ K < ∞, i.e., A−1n and Bn−1 exist and are uniformly bounded by some constant independent of n, then A−1n ∼ Bn−1.
5. If AnBn ∼ Cn and k A−1n k≤ K < ∞, then Bn ∼ A−1n Cn. Proof.
1. Eqs. (2.22) follows directly from (2.17).
2. |An− Cn| = |An− Bn+ Bn− Cn| ≤ |An− Bn| + |Bn− Cn|n−→→∞0 3. Applying lemma 2.2 yields
|AnCn− BnDn| = |AnCn− AnDn+ AnDn− BnDn|
≤ k Ank ·|Cn− Dn|+ k Dnk ·|An− Bn|
n−→→∞ 0.
4.
|A−1n − Bn−1| = |Bn−1BnAn− Bn−1AnA−1n
≤ k Bn−1 k · k A−1n k ·|Bn− An|
n−→→∞ 0.
2.3. ASYMPTOTICALLY EQUIVALENT MATRICES 11 5.
Bn− A−1n Cn = A−1n AnBn− A−1n Cn
≤ k A−1n k ·|AnBn− Cn|
n−→→∞ 0.
2 The above results will be useful in several of the later proofs.
Asymptotic equality of matrices will be shown to imply that eigenvalues, products, and inverses behave similarly. The following lemma provides a prelude of the type of result obtainable for eigenvalues and will itself serve as the essential part of the more general results to follow. It shows that if the weak norm of the difference of the two matrices is small, then the sums of the eigenvalues of each must be close.
Lemma 2.3 Given two matrices A and B with eigenvalues αn and βn, re-spectively, then
|n−1nX−1
k=0
αk− n−1nX−1
k=0
βk| = |n−1nX−1
k=0
(αk− βk)| ≤ |A − B|.
Proof: Define the difference matrix D = A− B = {dk,j} so that
nX−1 k=0
αk−nX−1
k=0
βk = Tr(A)− Tr(B)
= Tr(D).
Applying the Cauchy-Schwartz inequality (see, e.g., [19], p. 17) to Tr(D) yields
|Tr(D)|2 = ¯¯¯¯
¯
nX−1 k=0
dk,k¯¯¯¯
¯
2
≤ nnX−1
k=0
|dk,k|2
≤ nnX−1
k=0 nX−1 j=0
|dk,j|2
= n2|D|2. (2.23)
Taking the square root and dividing by n proves the lemma. 2 An immediate consequence of the lemma is the following corollary.
Corollary 2.2 Given two sequences of asymptotically equivalent matrices An and Bn with eigenvalues αn,k and βn,k, respectively, then
nlim→∞n−1 Dividing by n2, and taking the limit, results in
0≤ |n−1Tr(Dn)|2 ≤ |Dn|2 −→n→∞ 0 (2.27) from the lemma, which implies (2.26) and hence (2.24). 2 The previous corollary can be interpreted as saying the sample or arith-metic means of the eigenvalues of two matrices are asymptotically equal if the matrices are asymptotically equivalent. It is easy to see that if the matrices are Hermitian, a similar result holds for the means of the squared eigenvalues.
From (2.18) and (2.15),
|Dn| ≥ | |An| − |Bn| |
Corollary 2.3 Given two sequences of asymptotically equivalent Hermitian matrices An and Bn with eigenvalues αn,k and βn,k, respectively, then
nlim→∞n−1
2.3. ASYMPTOTICALLY EQUIVALENT MATRICES 13
or, equivalently,
nlim→∞n−1
nX−1 k=0
(α2n,k− βn,k2 ) = 0. (2.29)
Both corollaries relate limiting sample (arithmetic) averages of eigenval-ues or moments of an eigenvalue distribution rather than individual eigen-values. Equations (2.24) and (2.28) are special cases of the following funda-mental theorem of asymptotic eigenvalue distribution.
Theorem 2.2 Let An and Bn be asymptotically equivalent sequences of ma-trices with eigenvalues αn,k and βn,k, respectively. Assume that the eigenvalue moments of either matrix converge, e.g., lim
n→∞n−1
nX−1 k=0
αsn,k exists and is finite for any positive integer s. Then
nlim→∞n−1
nX−1 k=0
αsn,k = lim
n→∞n−1
nX−1 k=0
βn,ks . (2.30)
Proof. Let An = Bn+ Dn as in Corollary 2.2 and consider Asn− Bns
= ∆∆ n. Since the eigenvalues of Asn are αsn,k, (2.30) can be written in terms of ∆n as
nlim→∞n−1Tr∆n= 0. (2.31) The matrix ∆n is a sum of several terms each being a product of ∆0ns and Bn0s but containing at least one Dn. Repeated application of lemma 2.2 thus gives
|∆n| ≤ K0|Dn|n−→→∞0. (2.32) where K0does not depend on n. Equation (2.32) allows us to apply Corollary 2.2 to the matrices Asn and Dsn to obtain (2.31) and hence (2.30). 2 Theorem 2.2 is the fundamental theorem concerning asymptotic eigen-value behavior. Most of the succeeding results on eigeneigen-values will be appli-cations or specializations of (2.30).
Since (2.28) holds for any positive integer s we can add sums correspond-ing to different values of s to each side of (2.28). This observation immedi-ately yields the following corollary.
Corollary 2.4 Let An and Bn be asymptotically equivalent sequences of ma-trices with eigenvalues αn,k and βn,k, respectively, and let f (x) be any poly-nomial. Then
nlim→∞n−1
nX−1 k=0
f (αn,k) = lim
n→∞n−1
nX−1 k=0
f (βn,k) . (2.33) Whether or not An and Bn are Hermitian, Corollary 2.4 implies that (2.33) can hold for any analytic function f (x) since such functions can be expanded into complex Taylor series, i.e., into polynomials. If An and Bn are Hermitian, however, then a much stronger result is possible. In this case the eigenvalues of both matrices are real and we can invoke the Stone-Weierstrass approximation theorem ([19], p. 146) to immediately generalize Corollary 2.4. This theorem, our one real excursion into analysis, is stated below for reference.
Theorem 2.3 (Stone-Weierstrass) If F (x) is a continuous complex function on [a, b], there exists a sequence of polynomials pn(x) such that
nlim→∞pn(x) = F (x) uniformly on [a, b].
Stated simply, any continuous function defined on a real interval can be approximated arbitrarily closely by a polynomial. Applying theorem 2.3 to Corollary 2.4 immediately yields the following theorem:
Theorem 2.4 Let An and Bnbe asymptotically equivalent sequences of Her-mitian matrices with eigenvalues αn,k and βn,k, respectively. Since An and Bn are bounded there exist finite numbers m and M such that
m≤ αn,k, βn,k ≤ M , n = 1, 2, . . . k = 0, 1, . . . , n − 1. (2.34) Let F (x) be an arbitrary function continuous on [m, M ]. Then
nlim→∞n−1
nX−1 k=0
F (αn,k) = lim
n→∞n−1
nX−1 k=0
F (βn,k) (2.35) if either of the limits exists. Equivalently,
nlim→∞n−1
nX−1 k=0
(F (αn,k)− F (βn,k)) = 0 (2.36)
2.3. ASYMPTOTICALLY EQUIVALENT MATRICES 15 Theorem 2.4 is the matrix equivalent of theorem (7.4a) of [13]. When two real sequences{αn,k; k = 0, 1, . . . , n−1} and {βn,k; k = 0, 1, . . . , n−1} satisfy (2.34)-(2.35), they are said to be asymptotically equally distributed ([13], p.
62, where the definition is attributed to Weyl).
As an example of the use of theorem 2.4 we prove the following corollary on the determinants of asymptotically equivalent matrices.
Corollary 2.5 Let Anand Bnbe asymptotically equivalent Hermitian matri-ces with eigenvalues αn,k and βn,k, respectively, such that αn,k, βn,k ≥ m > 0.
from which (2.37) follows. 2
With suitable mathematical care the above corollary can be extended to cases where αn,k, βn,k > 0, but there is no m satisfying the hypothesis of the corollary, i.e., where the eigenvalues can get arbitrarily small but are still strictly positive. (In particular, see the discussion on p. 66 and in Section 3.1 of [13] for the required technical conditions.)
In the preceding chapter the concept of asymptotic equivalence of matri-ces was defined and its implications studied. The main consequenmatri-ces have been the behavior of inverses and products (theorem 2.1) and eigenvalues (theorems 2.2 and 2.4). These theorems do not concern individual entries in the matrices or individual eigenvalues, rather they describe an “average”
behavior. Thus saying A−1n ∼ Bn−1 means that that |A−1n − B−1n | n−→→∞ 0 and says nothing about convergence of individual entries in the matrix. In certain cases stronger results on a type of elementwise convergence are possible using the stronger norm of Baxter [1, 2]. Baxter’s results are beyond the scope of this book.