c 2004 Institute for Scientific
°
Computing and Information
INTERNATIONAL JOURNAL OF
INFORMATION AND SYSTEMS SCIENCES
Volume 1, Number 1, Pages 1–18
SOME ASYMPTOTIC BEHAVIORS ASSOCIATED WITH
MATRIX DECOMPOSITIONS
HUAJUN HUANG AND TIN-YAU TAM
(Communicated by Fuzhen Zhang)
Dedicated to Professor Yik-Hoi Au-Yeung on the occasion of his seventieth birthday
Abstract. We obtain several asymptotic results on the powers of a square
matrix associated with SVD, QR decomposition and Cholesky decomposition.
Key Words. a-component, d-component, SVD, QR decomposition, Cholesky
decomposition.
1. Yamamoto’s theorem
Given X ∈ Cn×n , the eigenvalue moduli of X are always less than or equal to
the spectral norm of X since
kXk := max kXvk2
kvk2 =1
and if Xv = λv for some unit vector v, then |λ| = kXvk2 ≤ kXk. So for all m ∈ N,
r(X) ≤ kX m k1/m ≤ kXk,
where r(X) denotes the spectral radius of X. The following is the celebrated
Beurling-Gelfand’s theorem (the finite dimensional case) [15, p.235, p.379].
Theorem 1.1. Let X ∈ Cn×n . then
lim kX m k1/m = r(X).
(1.1)
m→∞
Needless to say, (1.1) is true for all norms since they are all equivalent. We now
provide an elementary proof which is different from those in [24, 13, 7, 16].
Proof. Since k · k is invariant under unitary similarity, by Schur triangularization
theorem, we may assume that X = T is upper triangular with ascending diagonal
moduli |t11 | ≤ · · · ≤ |tnn |. When X is nilpotent, that is, r(X) = 0, (1.1) is obviously
true. Hence we may assume that X is not nilpotent so that r(X) = |tnn | 6= 0. Write
(m)
T m = [tij ] ∈ Cn×n which is also upper triangular. For 1 ≤ i ≤ j ≤ n,
(m)
tij
(1.2)
=
X
m
Y
i=p0 ≤p1 ≤···≤pm =j
`=1
(m)
tp`−1 p` .
Clearly tii = tm
ii , i = 1, . . . , n. Since both sides of (1.1) are homogenous, by
(m)
appropriate scaling, we may assume that |tnn | ≥ 1. Let us estimate |tij | for fixed
i < j. The number of (m + 1)-tuples (p0 , p1 , · · · , pm ), where i = p0 ≤ p1 ≤ · · · ≤
2000 Mathematics Subject Classification. 15A23, 15A42.
1
2
H. HUANG AND T.Y. TAM
¡
¢
pm = j are integers, is equal to j−i+m−1
. For each of such (p0 , · · · , pm ), there are
m−1
at most j − i numbers `’s in {1, · · · , m} such that p`−1 6= p` . Let
c := max |tpq | ≥ 1
1≤p,q≤n
denote the maximal entry modulus of T . By (1.2) and the fact that |t11 | ≤ · · · ≤
|tnn |, when m ≥ n,
(m)
|tij |
X
m
Y
i=p0 ≤p1 ≤···≤pm =j
`=1
≤
X
≤
|tp`−1 p` |
m−j+i
cj−i |tjj |
i=p0 ≤p1 ≤···≤pm =j
µ
¶
j − i + m − 1 j−i
c |tjj |m−j+i
m−1
µ
¶
n + m − 2 n−1
≤
c
|tnn |m .
m−1
=
(1.3)
By (1.3), for m ≥ n,
|tnn |m ≤ kT m k ≤
j
n X
X
µ
2 n−1
|tm
ij | ≤ n c
j=1 i=1
¶
n+m−2
|tnn |m .
m−1
Taking m-th roots on all sides and taking limits for m → ∞ lead to
lim kT m k1/m = |tnn | = r(T ).
m→∞
¤
Suppose that the singular values s1 (X), . . . , sn (X) of X and the eigenvalues
λ1 (X), . . . , λn (X) of X are arranged in descending order
s1 (X) ≥ s2 (X) ≥ · · · ≥ sn (X),
|λ1 (X)| ≥ |λ2 (X)| ≥ · · · ≥ |λn (X)|.
Since kXk = s1 (X) and r(X) = |λ1 (X)|, the following result of Yamamoto [24] is
a direct generalization of (1.1):
lim [si (X m )]1/m = |λi (X)|,
(1.4)
m→∞
i = 1, . . . , n.
We now provide a slight extension of Yamamoto’s result and the proof makes
use of compound matrix like Yamamoto’s original proof.
Theorem 1.2. Let A, B, X ∈ Cn×n such that A, B are nonsingular. Then
(1.5)
lim [sk (AX m B)]1/m = |λk (X)|,
m→∞
k = 1, . . . , n.
Proof. It suffices to prove (1.5) for B = In since AX m B = AB(B −1 XB)m and the
spectrum of X and B −1 XB are identical. We first establish the case k = 1. Since
k · k is submultiplicative, for all m ∈ N,
1
kA−1 k1/m
kX m k1/m ≤ kAX m k1/m ≤ kAk1/m kX m k1/m .
Since kA−1 k1/m and kAk1/m converge to 1,
(1.6)
is established by (1.1).
lim [s1 (AX m )]1/m = |λ1 (X)|
m→∞
ASYMPTOTIC BEHAVIOR
3
Let Ck (X) denote the k-th compound of X. It is well-known that
(1.7)
|λ1 (Ck (X))| =
k
Y
|λi (X)|,
s1 (Ck (X)) =
i=1
k
Y
si (X).
i=1
So by (1.6),
(1.8)
lim [
m→∞
k
Y
si (AX m )]1/m =
i=1
k
Y
|λi (X)|.
i=1
When X is nilpotent, that is, |λ1 (X)| = 0, (1.5) is obviously true. We now assume
that X is not nilpotent. Let X have k nonzero eigenvalues for some 1 ≤ k ≤ n,
that is,
|λ1 (X)| ≥ · · · ≥ |λk (X)| > |λk+1 (X)| = · · · = |λn (X)| = 0.
Qt
By (1.8) for sufficiently large m, we have i=1 si (AX m ) > 0 when 1 ≤ t ≤ k. By
(1.8), for 1 ≤ j ≤ k + 1
"Q
#1/m
j
m
s
(AX
)
i
i=1
lim [sj (AX m )]1/m = lim Qj−1
m
m→∞
m→∞
i=1 si (AX )
hQ
i1/m
j
lim
si (AX m )
i=1
= m→∞ h
i1/m
Qj−1
lim
si (AX m )
i=1
m→∞
Qj
i=1 |λi (X)|
= Qj−1
= |λj (X)|.
i=1 |λi (X)|
1
In particular limm→∞ [sk+1 (AX m )] m = 0 and the proof is completed.
¤
Remark 1.3. The nonsingularity of A and B is clearly needed. For example if
·
¸
·
¸
·
¸
0 0
1 0
1 2
A=
, B=
, X=
,
0 1
0 0
0 3
then r(X) = 3 but AX m B = 0 for all m ∈ N.
Remark 1.4. One may use the inequality [8, p.178] si (AB) ≤ s1 (A)si (B), A, B ∈
Cn×n and continuity argument to have
sn (A)si (X m ) ≤ si (AX m ) ≤ s1 (A)si (X m ),
i = 1, . . . , n.
Then by (1.4) we arrive at (1.5).
Remark 1.5. Yamamoto’s theorem (1.4) was extended in the context of semisimple Lie group [22]. It involves Cartan decomposition and complete multiplicative
Jordan decomposition.
2. Asymptotic result for QR decomposition
Let {e1 , . . . , en } be the standard basis of Cn , that is, ei has 1 as the only nonzero
entry at the i-th position. We identify a permutation ω ∈ Sn with the unique
permutation matrix (also written as ω) in the general linear group GLn (C), where
ωei = eω(i) . The matrix representation of ω under the standard basis is
£
¤
ω = eω(1) , · · · , eω(n) .
4
H. HUANG AND T.Y. TAM
Given X ∈ GLn (C), the well-known QR decomposition asserts that X = QR,
where Q is unitary and R is upper triangular with positive diagonal entries. The
decomposition is unique. Denote
a(X) := diag R,
where diag A ∈ Cn×n denotes the diagonal matrix whose diagonal is that of A ∈
Cn×n . When v ∈ Cn×n , diag v means the diagonal matrix with diagonal v. Recently
it was shown in [9] that given A, B ∈ GLn (C), the limit
lim [a(AX m B)]1/m
m→∞
exists and is related to the eigenvalue moduli of X. More precisely,
Theorem 2.1. [9] Let A, B, X ∈ GLn (C). Let X = Y −1 JY be the Jordan decomposition of X, where J is the Jordan form of X, diag J = diag (λ1 , . . . , λn )
satisfying |λ1 | ≥ · · · ≥ |λn |. Then
lim [a(AX m B)]1/m = diag (|λω(1) |, . . . , |λω(n) |),
m→∞
where the permutation ω is uniquely determined by the LωU decomposition of
Y B = LωU , where L is unit lower triangular and U is upper triangular.
The quantity
a(X) = diag R
has a very nice geometric meaning: ai (X) is the distance (in 2-norm) between the
i-th column of X and the span of the previous i − 1 columns, i = 1, . . . , n (we adopt
the convention that the span of a empty set is the zero space). Since all norms of
Cn are equivalent, we have
Corollary 2.2. Let k · k be a norm on Cn and i = 1, . . . , n. The m-th root of the
distance in k · k between the i-th column of AX m B and the span of the previous
i − 1 columns of AX m B converges to |λω(i) |.
The LωU decomposition is known as Gelfand-Naimark decomposition [5, p.434].
Remark 2.3. There are some interesting features of the result.
(1) The limit is independent of A but depends on B.
(2) The permutation matrix ω is not unique since Y is not unique. However
the limit remains the same.
See [9] for explanations.
Remark 2.4. One may want to consider singular X. However the QR decomposition of a singular X ∈ Cn×n is not unique. Indeed the a-component of X is not
unique either. For example:

 


1 0 0
1 0 0
1 0 0
0 1 = 0 0 1
0 0
X := 
0
0 1 0
1
Remark 2.5. Theorem 2.1 was extended in the context of semisimple Lie group
[10] which involves Iwasawa decomposition, complete multiplicative Jordan decomposition and Bruhat decomposition.
ASYMPTOTIC BEHAVIOR
5
Remark 2.6. One may not replace the Jordan decomposition in Theorem 2.1 by
Schur (upper) triangularization decomposition (with descending diagonal moduli)
to get ω. For example, consider


3 1 1
X :=  0 2 0  .
0 2 3
Then

X = V −1 T V,

3 1 1
3 2,
T =
2

1
V = 0
0
0
0
1

0
1
0
is the Schur (upper) triangularization decomposition of X, where T is a Schur
triangular form of X, with descending diagonal moduli. Clearly V = I3 ·V ·I3 is the
Gelfand-Naimark decomposition, that is, ωV of V is the transposition (23) so that
λωV := (λωV (1) , λωV (2) , λωV (3) ) = (3, 2, 3). The following is a Jordan decomposition
of X:




3 1 0
0.5 0 0.25
X = Y −1 JY, J =  0 3 0  , Y =  0 1 0.5  ,
0 0 2
0 1
0
and the Gelfand-Naimark decomposition of Y = LωY U is given as




1 0 0
0.5 0 0.25
L =  0 1 0  , U =  0 1 0.5 
0 1 1
0 0 −0.5
and ωY = I3 so that λωY = (3, 3, 2).
The above example fails because of a “bad” choice of Schur decomposition as we
shall see in Theorem 2.7. There is a variant of Gelfand-Naimark decomposition.
Each matrix Y ∈ GLn (C) can be written as
Y = V ωU
where ω is a permutation matrix, V is unit upper triangular and U is upper triangular. The permutation ω is unique in the decomposition. It can easily be deduced
from the Gelfand-Naimark decomposition because if


1


1


K := 
 ∈ Cn×n ,
.
.


.
1
then the Gelfand-Naimark decomposition for KY yields KY = Lω 0 U . So Y =
KLK(Kω 0 )U since K −1 = K. Notice that V := KLK is unit upper triangular.
Set ω = Kω 0 .
We have the following theorem.
Theorem 2.7. Suppose X = Y −1 T Y ∈ GLn (C), where T ∈ Cn×n is an upper
triangular matrix, and the diagonal entries of T satisfy that |t11 | ≤ |t22 | ≤ · · · ≤
|tnn |, so that λk := tkk are the eigenvalues of X with |λ1 | ≤ |λ2 | ≤ · · · ≤ |λn |. Then
(2.1)
lim [a(AX m B)]1/m = diag (|λω(1) |, . . . , |λω(n) |),
m→∞
where ω is a permutation determined by the decomposition Y B = V ωU of Y B,
such that V is unit upper triangular and U is upper triangular.
6
H. HUANG AND T.Y. TAM
Corollary 2.8. Suppose X = Y −1 T Y ∈ GLn (C) where T ∈ Cn×n is a lower
triangular matrix and the diagonal entries of T satisfy that |t11 | ≥ |t22 | ≥ · · · ≥
|tnn |, so that λk := tkk are the eigenvalues of X with |λ1 | ≥ |λ2 | ≥ · · · ≥ |λn |. Then
lim [a(AX m B)]1/m = diag (|λω(1) |, . . . , |λω(n) |),
m→∞
where ω is a permutation determined by the Gelfand-Naimark decomposition Y B =
LωU of Y B, such that L is unit lower triangular and U is upper triangular.
Proof. Notice that X = Y −1 T Y = Y −1 K(KT K)KY , where KT K is upper triangular with ascending diagonal moduli |µ1 | ≤ · · · ≤ |µn | and µk := λn−k+1 ,
k = 1, . . . , n. Moreover KY B = (KLK)KωU in which KLK is unit upper triangular. Apply Theorem 2.7 to have
lim [a(AX m B)]1/m = diag (|µKω(1) |, . . . , |µKω(n) |) = diag (|λω(1) |, . . . , |λω(n) |).
m→∞
¤
To prove Theorem 2.7 we need two lemmas on compound matrices. Suppose
1 ≤ k ≤ n and let
Qk,n := {ω = (ω(1), . . . , ω(k)) : 1 ≤ ω(1) < · · · < ω(k) ≤ n}
be the set of strictly increasing¡ sequences
of length k chosen from 1, . . . , n. The
¢
number of elements in Qk,n is nk . The elements of Qk,n are ordered by the lexicographic order ¹ (a total order) and traditionally the
¡ ¢ entries of the compound
matrix Ck (X) are indexed by Qk,n . Let σ : {1, 2, . . . , nk } → Qk,n be the natural
bijection such that i < j if and only if σ(i) ≺ σ(j). Let ω E γ denote the partial
order in Qk,n : ω(t) ≤ γ(t) for all t = 1, . . . , k.
Lemma 2.9. Let T ∈£Cn×n
¤ be an upper triangular matrix. Then the kth compound Ck (T ) =: T 0 = t0ij ∈ C(n)×(n) is an upper triangular matrix. Moreover,
t0ij 6= 0 only if σ(i) E σ(j).
k
k
Proof. One has t0ij = det T [σ(i)|σ(j)], where T [σ(i)|σ(j)] denotes the submatrix
of T by choosing the rows of T corresponding to σ(i) ∈ Qk,n and the columns
of T corresponding to σ(j) ∈ Qk,n . ¡It¢suffices to prove that if σ(i) 5 σ(j) then
det T [σ(i)|σ(j)] = 0 for i, j ∈ {1, · · · , nk }.
Denote σi := σ(i) and σj := σ(j). Then σi 5 σj means that there is t ∈
{1, · · · , k} such that σi (t) > σj (t). So
σi (k) > σi (k − 1) > · · · > σi (t) > σj (t) > · · · > σj (2) > σj (1).
The lower left (k + 1 − t) × t submatrix of T [σi |σj ] is also the submatrix of T
corresponding to the σi (t), · · · , σi (k) rows and the σj (1), · · · , σj (t) columns of T ,
which is a zero matrix. So the first t columns of T [σi |σj ] span a space of dimension
no more than k − (k + 1 − t) = t − 1. Therefore, det T [σi |σj ] = 0.
¤
Lemma 2.10. Suppose that T ∈ Cn×n is upper triangular with
diagonal
¡ ¢ ascending
¡ ¢
moduli |t11 | ≤ · · · ≤ |tnn |. Fix 1 ≤ k ≤ n. Then there is an nk × nk permutation
matrix P (depending on |t11 |, · · · , |tnn |) such that P −1 Ck (T )P is upper triangular
with ascending diagonal moduli and P −1 Ck (S)P is upper triangular for all upper
triangular matrices S ∈ Cn×n .
£ ¤
Proof. By Lemma 2.9, Ck (T ) =: T 0 = t0ij ∈ C(n)×(n) is a upper triangular
k
k
matrix, with t0ij = det T [σ(i)|σ(j)]. Let P be the permutation of shortest length
(that is, smallest number of transposition in its transposition decomposition) such
ASYMPTOTIC BEHAVIOR
7
¡ ¢
that the diagonal moduli of P −1 Ck (T )P , namely, |t0P (i)P (i) |, i = 1, . . . , nk , are
in ascending order. Then P rearranges t011 , · · · , t0 n n in the way that it moves
(k )(k )
the smallest modulus term(s) to the beginning position(s), then moves the second
smallest modulus term(s) to the consecutive position(s), and then moves the third
smallest modulus term(s), and so on. Moreover, the fact that P has the shortest
possible length among all such rearrangement(s) implies that if the i-th and the
j-th diagonal terms of P −1 Ck (T )P (namely t0P (i)P (i) and t0P (j)P (j) ) have the same
modulus for certain i > j, then the positions of these two terms in the diagonal of
Ck (T ) (namely the P (i)-th and P (j)-th positions) satisfy P (i) > P (j).
With these facts in mind, we claim that if S ∈ Cn×n is upper triangular, then
P −1 Ck (S)P ∈ C(n)×(n) is also upper triangular. Now Ck (S) = [s0ij ] ∈ C(n)×(n)
k
k
k
k
¡ ¢
where s0ij = det S[σ(i)|σ(j)] for i, j ∈ {1, · · · , nk }, and
P −1 Ck (S)P = [s0P (i)P (j) ] ∈ C(n)×(n) .
k
k
¡ ¢
If P Ck (S)P were not upper triangular, then we would have i > j in {1, · · · , nk }
such that s0P (i)P (j) 6= 0. So σ(P (i)) E σ(P (j)) by Lemma 2.9. On one hand, we
have σ(P (i)) ¹ σ(P (j)) and thus P (i) ≤ P (j). On the other hand,
Y
Y
|t0P (i)P (i) | = | det T [σ(P (i))|σ(P (i))]| =
|trr | ≤
|trr | = |t0P (j)P (j) |
−1
r∈σ(P (i))
r∈σ(P (j))
which would not occur because
(1) it is impossible that |t0P (i)P (i) | < |t0P (j)P (j) | since i > j and the diagonal
¡ ¢
moduli of P −1 Ck (T )P , namely, |t0P (i)P (i) |, i = 1, . . . , nk , are in ascending
order;
(2) it is also impossible that |t0P (i)P (i) | = |t0P (j)P (j) | since this together with
i > j and P (i) ≤ P (j) contradicts the fact that P has the shortest possible
length and the discussion at the end of the preceding paragraph.
This means that P −1 Ck (S)P ∈ C(n)×(n) is upper triangular. In particular, P −1 Ck (T )P ∈
k
k
¤
C(n)×(n) is upper triangular.
k
k
We now prove Theorem 2.7.
Proof. It suffices to prove (2.1) for B = In since AX m B = AB(B −1 XB)m and the
(m)
spectrum of X and B −1 XB are identical. Denote ak := ak (AX m ), k = 1, . . . , n.
We will first prove that
(m)
lim [a1 ]1/m = |λω(1) |
(2.2)
m→∞
(m)
and then use compound matrix. Notice that a1
1
k(AY
−1 )−1 k
= kAY −1 T m Y e1 k2 and
kT m Y e1 k2 ≤ kAY −1 T m Y e1 k2 ≤ kAY −1 kkT m Y e1 k2
so (2.2) amounts to
(2.3)
1/m
lim kT m Y e1 k2
m→∞
= |λω(1) |.
Since Y = V ωU ,
kT m Y e1 k2 = kT m V ωU e1 k2 = |u11 |kT m vω(1) k2
8
H. HUANG AND T.Y. TAM
where vi denotes the i-th column of V . Since |u11 | > 0 is a constant, it suffices to
show
1/m
lim kT m vω(1) k2
(2.4)
m→∞
= |λω(1) |.
Since both sides of (2.4) are homogenous, by an appropriate scaling, we may assume
(m)
|λi | ≥ 1 for all i = 1, . . . , n. To this end, write T m = [tij ] ∈ Cn×n which is upper
triangular since T is upper triangular.
(m)
Let tj denote the j-th column of T m . Since V is unit upper triangular, the
vector T m vω(1) is a linear combination (with constant coefficients) of the first ω(1)
columns of T m , and the coefficient for the ω(1)-th column of T m in the linear
combination is 1. On one hand, we have
|λω(1) |m ≤ kT m vω(1) k2 .
(2.5)
On the other hand
ω(1)
kT m vω(1) k2 = k
X
ω(1)
ω(1)
(m)
tj
vjω(1) k2 ≤
j=1
X
(m)
|vjω(1) | ktj
k2 ≤ bω(1)
X
(m)
ktj
k2 ,
j=1
j=1
where bω(1) := maxj=1,...,ω(1) |vjω(1) | ≥ 1 is a constant. Notice that
(m)
ktj k2
(2.6)
≤
(m)
ktj k1
=
j
X
(m)
|tij |.
i=1
When i ≤ j and m ≥ n, by |t11 | ≤ · · · ≤ |tnn | and (1.3),
µ
¶
µ
¶
j − i + m − 1 j−i
n + m − 2 n−1
(m)
m−j+i
(2.7)
|tij | ≤
c |λj |
≤
c
|λj |m ,
m−1
m−1
where c := max1≤p,q≤n |tpq | ≥ 1. Hence from (2.6) and (2.7)
ω(1) j µ
X X n + m − 2¶
cn−1 |λj |m
kT m vω(1) k2 ≤ bω(1)
m
−
1
j=1 i=1
µ
¶
n+m−2
2
(2.8)
≤
bω(1) ω(1) cn−1 |λω(1) |m .
m−1
So from (2.5) and (2.8),
1/m
lim kT m vω(1) k2
m→∞
= |λω(1) |,
or equivalently, (2.2) is established. If AX m = Qm Rm is the QR decomposition of
AX m , then Ck (AX m ) = Ck (Qm )Ck (Rm ) is the QR decomposition of Ck (AX m ) so
that
k
k
Y
Y
(m)
m
m
a1 (Ck (A)Ck (X) ) =
ai (AX ) =
ai .
i=1
i=1
Though Ck (X) = Ck (Y )−1 Ck (T )Ck (Y ) and Ck (T ) is upper triangular, we are not
quite ready to apply (2.2) on
Ck (A)Ck (X)m = Ck (AX m ).
It is because the diagonal moduli of C¡k (T
¢ ) are
¡ ¢not necessarily in ascending order.
However by Lemma 2.10 there is an nk × nk permutation matrix P such that
P −1 Ck (T )P is upper triangular and diag (P −1 Ck (T )P ) has descending diagonal
moduli and P −1 Ck (V )P is still unit upper triangular. Now
Ck (X) = (P −1 Ck (Y ))−1 (P −1 Ck (T )P )(P −1 Ck (Y ))
ASYMPTOTIC BEHAVIOR
9
and
P −1 Ck (Y ) = (P −1 Ck (V )P )(P −1 Ck (ω))Ck (U ).
The (1, 1) entry of
(P −1 Ck (ω))−1 [diag (P −1 Ck (T )P )](P −1 Ck (ω))
= Ck (ω)−1 P (P −1 [diag Ck (T )]P )P −1 Ck (ω)
= Ck (ω)−1 Ck (diag T )Ck (ω)
= Ck (ω −1 (diag T )ω)
is
Qk
i=1
λω(i) . By (2.2), for 1 ≤ k ≤ n,
lim
m→∞
k
Y
(m) 1/m
|ai
|
= lim (a1 (Ck (A)Ck (X)m )1/m =
m→∞
i=1
Qk
=
(m) 1/m
|
i=1 |ai
lim Qk−1
(m) 1/m
m→∞
|
i=1 |ai
Qk
=
i=1
|λω(i) |
i=1
|λω(i) |
Qk−1
|λω(i) |.
i=1
So
(m)
lim |a |1/m
m→∞ k
k
Y
lim
=
m→∞
lim
m→∞
Qk
i=1
(m) 1/m
|ai
|
Qk−1
(m) 1/m
|
i=1 |ai
= |λω(k) |.
This completes the proof.
¤
3. Cholesky component
Given a positive definite matrix P ∈ Cn×n , Cholesky decomposition asserts that
P = R∗ R,
where R ∈ Cn×n is an upper triangular matrix with positive diagonal entries and
the decomposition is unique. Let
d(P ) = (diag R)2 .
The spectral theorem asserts that there is a unitary matrix Y ∈ Cn×n such that
P = Y −1 ΛY
where Λ := diag (λ1 , . . . , λn ) with descending positive diagonal entries. So the
power P m remains positive definite and thus d(P m ) is well defined.
Theorem 3.1. Let P ∈ Cn×n be positive definite. Let P = Y −1 ΛY where Y ∈
Cn×n is unitary and Λ = diag (λ1 , . . . , λn ), where λ1 ≥ · · · ≥ λn > 0. Then
(3.1)
lim d(P m )1/m = diag (λω(1) , . . . , λω(n) )
m→∞
where ω is the permutation uniquely determined by the Gelfand-Naimark decomposition Y = LωU .
Proof. Notice that P 1/2 = Y −1 Λ1/2 Y . Let P m/2 = Qm Rm be the QR decomposition of P m/2 . Then
∗
P m = P m/2 P m/2 = (P m/2 )∗ P m/2 = Rm
Rm .
So
(3.2)
d(P m ) = [a(P m/2 )]2
10
H. HUANG AND T.Y. TAM
and thus
lim d(P m )1/m
m→∞
lim [a(P m/2 )]2/m
=
m→∞
( lim a([P 1/2 ]m )1/m )2
=
m→∞
1/2
1/2
= [diag (λω(1) , . . . , λω(n) )]2
= diag (λω(1) , . . . , λω(n) )
by Theorem 2.1 or Corollary 2.8.
¤
Remark 3.2. When P is positive semidefinite, neither the Cholesky decomposition
of P nor d(P ) is unique. For example
¸
¸
·
·
0 0
0 1
P :=
= P ∗ P = R∗ R, R =
.
0 1
0 0
We conclude our discussion with a result on the d-component of the set of positive definite matrices with prescribed eigenvalues. Kostant’s nonlinear convexity
theorem [11, Theorem 4.1] asserts that if Λ = diag (λ1 , . . . , λn ), where λ1 , . . . , λn
are positive, then
{a(ΛY ) : Y ∈ Cn×n is unitary} = exp(conv Sn log Λ),
(3.3)
where Sn is the full symmetric group on {1, . . . , n}, Sn x := {(xω(1) , . . . , xω(n) ) :
ω ∈ Sn }, and conv denotes the convex hull of the underlying set.
Theorem 3.3. Let Λ = diag (λ1 , . . . , λn ), where λ1 , . . . , λn are positive, and
O(Λ) := {Y −1 ΛY : Y ∈ Cn×n is unitary}. Then
d(O(Λ)) = exp(conv Sn log Λ),
where log λ = diag (log λ1 , . . . , log λn ).
Proof.
d(O(Λ))
= {d(Y −1 ΛY ) : Y unitary}
= {[a(Y −1 Λ1/2 Y )]2 : Y unitary}
= {[a(Λ1/2 Y )]2 : Y unitary}
= [exp(conv Sn log Λ1/2 )]2
= exp(conv Sn log Λ).
by (3.3).
¤
Remark 3.4. In terms of inequalities d ∈ exp(conv Sn log Λ) means multiplicative
majorization
max
σ∈Sn
and
Qn
i=1
di =
Qn
i=1
k
Y
i=1
dσ (i) ≤ max
σ∈Sn
k
Y
λσ(i) ,
k = 1, . . . , n − 1,
i=1
λn .
Remark 3.5. Theorem 3.1 and Theorem 3.3 can be extended to semisimple Lie
group along the same reasoning [10]. For generalized Cholesky decomposition see
[5, p.272-273].
ASYMPTOTIC BEHAVIOR
11
Remark 3.6. In [11] two convexity theorems are proved. One is the above Kostant’s
nonlinear convexity theorem (3.3) concerning the a-component of nonsingular A ∈
Cn×n (more generally a-component of an element with respect to Iwasawa decomposition in real semisimple Lie group). Notice that (3.3) may be written as
(3.4)
{a(Y1 ΛY2 ) : Y1 , Y2 ∈ Cn×n are unitary} = exp(conv Sn log Λ),
because Y1 on the left would not change the a-component. Denote by
Π(Λ) := {Y1 ΛY2 : Y1 , Y2 ∈ Cn×n are unitary}
the set of nonsingular matrices in Cn×n with fixed singular values λ1 , . . . , λn (> 0).
Then (3.4) and thus (3.3) amount to
(3.5)
a(Π(Λ)) = exp(conv Sn log Λ).
Certainly the set conv Sn log Λ is convex. It is called nonlinear convexity theorem
because the action X 7→ a(X) is nonlinear.
The other convexity theorem [11, Theorem 8.2] is known as Kostant’s linear
convexity theorem. It generalizes the well-known Schur-Horn’s result [16, 6] in
the context of real semisimple Lie algebra. Let Λ := diag (λ1 , . . . , λn ), where
λ1 · · · , λn ∈ R. Let
Γ(Λ) := {Y −1 ΛY : Y ∈ Cn×n is unitary}
be the orbit of n × n Hermitian matrices with fixed eigenvalues λ’s. Schur-Horn’s
result asserts that
(3.6)
diag (Γ(Λ)) = conv Sn Λ.
It is called linear convexity theorem because the projection X 7→ diag X is a linear
map on Cn×n . The original statement of Horn [6, Theorem 4] asserts that
(3.7)
diag (Γ(Λ)) = {Dλ : D is doubly stochastic},
where λ = (λ1 , . . . , λn )T (Horn established the inclusion ⊃ and the other inclusion
is obvious since orthostochastic matrices are doubly stochastic). From Birkhoff’s
theorem [7] one can see (3.6) and (3.7) are equivalent.
Duistermaat [1] showed that the Kostant’s linear and nonlinear convexity theorems can be deduced from each other. Also see an earlier work of Heckman [6].
Both proofs of Duistermaat and Heckman use homotopy argument. Duistermaat’s
approach shows the existence of an analytic map satisfying some nice properties
and the proof is highly analytic. In particular (3.3) and (3.6) are equivalent.
See [18] to see how to deduce majorization inequalities [7] from (3.6) via the root
system. The inequalities are indeed the description of the hyperplanes defining
the convex polytope conv Sn Λ. The traditional approach of deriving inequalities
is via Hardy-Littlewood-Pólya theorem [4] and (3.6). Nevertheless we view (3.5)
and (3.6) more geometric than majorization or log-majorization. Also see [19] for
the explanation via Kostant’s linear convexity theorem (applying on the simple Lie
algebras son,n and sun,n ) of Thompson’s famous inequalities (on the diagonal entries
and singular values) involving a subtract term [23], that is, from a Lie theoretic
point of view.
The proof of Theorem 3.3 makes use of (3.3).
See [2, 17] for connection between majorization and generalized numerical range
and see [12, 20, 21] for more convexity results on generalized numerical range.
12
H. HUANG AND T.Y. TAM
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Department of Mathematics and Statistics, Auburn University, AL 36849–5310, USA
E-mail: [email protected], [email protected]
URL: http://www.auburn.edu/∼huanghu, http://www.auburn.edu/∼tamtiny