vol20_pp303-313. 117KB Jun 04 2011 12:06:08 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 20, pp. 303-313, May 2010
ELA
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PERTURBATIONS OF FUNCTIONS OF
DIAGONALIZABLE MATRICES∗
MICHAEL I. GIL’†
˜ be n × n diagonalizable matrices and f be a function defined on their
Abstract. Let A and A
˜ are established. Applications to
spectra. In the present paper, bounds for the norm of f (A) − f (A)
differential equations are also discussed.
Key words. Matrix valued functions, Perturbations, Similarity of matrices, Diagonalizable
matrices.
AMS subject classifications. 15A54, 15A45, 15A60.
n
1. Introduction and statement of the main result.
p Let C be a Euclidean
space with the scalar product (·, ·), Euclidean norm k·k = (·, ·) and identity operator
˜j (j = 1, . . . , n), respectively.
I. A and A˜ are n × n matrices with eigenvalues λj and λ
σ(A) denotes the spectrum of A, A∗ is the adjoint to A, and N2 (A) is the HilbertSchmidt (Frobenius) norm of A: N22 (A) = Trace(A∗ A).
In the sequel, it is assumed that each of the matrices A and A˜ has n linearly
independent eigenvectors, and therefore, these matrices are diagonalizable. In other
words, the eigenvalues of these matrices are semi-simple.
˜ The aim of this paper is to
Let f be a scalar function defined on σ(A) ∪ σ(A).
˜
establish inequalities for the norm of f (A) − f (A). The literature on perturbations
of matrix valued functions is very rich but mainly, perturbations of matrix functions
of a complex argument and matrix functions of Hermitian matrices were considered,
cf. [1, 11, 13, 14, 16, 18]. The matrix valued functions of a non-Hermitian argument
have been investigated essentially less, although they are very important for various
applications; see the book [10].
The following quantity plays an essential role hereafter:
g(A) :=
"
N22 (A)
−
n
X
k=1
|λk |
2
#1/2
.
∗ Received by the editors December 8, 2009. Accepted for publication April 19, 2010. Handling
Editor: Peter Lancaster.
† Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva
84105, Israel ([email protected]). Supported by the Kameah fund of the Israel.
303
Electronic Journal of Linear Algebra ISSN 1081-3810
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M.I. Gil’
g(A) enjoys the following properties:
g 2 (A) ≤ 2N22 (AI ) (AI = (A − A∗ )/2i) and g 2 (A) ≤ N22 (A) − |Trace(A2 )|, (1.1)
cf. [8, Section 2.1]. If A is normal, then g(A) = 0. Denote by µj , j = 1, . . . , m ≤ n,
the distinct eigenvalues of A, and by pj the algebraic multiplicity of µj . In particular,
˜j ,
one can write µ1 = λ1 = · · · = λp1 , µ2 = λp1 +1 = · · · = λp1 +p2 , etc. Similarly, µ
˜ and p˜j denotes the algebraic
j = 1, . . . , m
˜ ≤ n, are the distinct eigenvalues of A,
multiplicity of µ
˜j .
Let δj be the half-distance from µj to the other eigenvalues of A, namely,
δj :=
min
|µj − µk |/2 > 0.
min
|˜
µj − µ
˜k |/2 > 0.
k=1,...,m; k6=j
Similarly,
δ˜j :=
k=1,...,m;
˜ k6=j
Put
β(A) :=
m
X
j=1
pj
n−1
X
k=0
g k (A)
√
δjk k!
˜ :=
and β(A)
m
˜
X
p˜j
j=1
n−1
X
k=0
˜
g k (A)
√ .
δ˜k k!
j
Here g 0 (A) = δj0 = 1 and g˜0 (A) = δ˜j0 = 1. According to (1.1),
√
n−1
m
X ( 2N2 (AI ))k
X
√
pj
β(A) ≤
.
δjk k!
j=1
k=0
In what follows, we put
˜j )
f (λk ) − f (λ
˜j .
= 0 if λk = λ
˜j
λk − λ
Now we are in a position to formulate our main result.
Theorem 1.1. Let A and A˜ be n × n diagonalizable matrices and f be a function
˜ Then the inequalities
defined on σ(A) ∪ σ(A).
f (λ ) − f (λ
˜ j )
k
˜
˜
˜
N2 (f (A) − f (A)) ≤ β(A)β(A) max
(1.2)
N2 (A − A)
˜j
j,k
λk − λ
and
˜ j )|
˜ ≤ β(A)β(A)
˜ max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
(1.3)
are valid.
The proof of this theorem is divided into lemmas which are presented in the next
two sections. The importance of Theorem 1.1 lies in the fact that the right-hand sides
˜
of inequalities (1.2) and (1.3) only involve universal quantities calculated for A and A,
˜
and the values of the function f on the spectra σ(A) and σ(A), but e.g. no matrices
performing similarities of A and A˜ to diagonal ones.
Electronic Journal of Linear Algebra ISSN 1081-3810
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Perturbations of Functions of Diagonalizable Matrices
305
2. The basic lemma. Let T and T˜ be the invertible matrices performing similarities of A and A˜ to diagonal ones, that is,
T −1 AT = S,
(2.1)
˜
T˜ −1 A˜T˜ = S,
(2.2)
and
˜ j ). Everywhere below, {dk }n is the standard
where S = diag (λj ) and S˜ = diag (λ
k=1
orthonormal basis and kBk = sup {kBhk/khk : h ∈ Cn } for an n × n matrix B.
Recall that N2 (AB) ≤ kBkN2 (A).
We begin with the following lemma.
Lemma 2.1. Let the hypotheses of Theorem 1.1 hold. Then
˜ ≤ kT˜ −1k2 kT k2
N22 (f (A) − f (A))
n
X
j,k=1
˜ j ))(T −1 T˜dj , dk )|2 .
|(f (λk ) − f (λ
(2.3)
Proof. We have
˜ = N 2 (T ST −1 − T˜S˜T˜ −1 ) = N 2 (T ST −1T˜T˜ −1 − T T −1T˜S˜T˜ −1 )
N22 (A − A)
2
2
˜ T˜ −1 ) ≤ kT˜ −1 k2 kT k2N 2 (ST −1 T˜ − T −1 T˜S).
˜
= N22 (T (ST −1T˜ − T −1 T˜S)
2
But
˜ =
N22 (ST −1 T˜ − T −1 T˜S)
=
n
X
j,k=1
n
X
j,k=1
˜ j , dk )|2
|(ST −1 T˜ − T −1 T˜S)d
˜ j , dk )|2 = J,
|(T −1 T˜dj , S ∗ dk ) − (T −1 T˜Sd
where
J :=
n
X
j,k=1
˜j )(T −1 T˜dj , dk )|2 .
|(λk − λ
So,
˜ ≤ kT˜ −1k2 kT k2 J.
N22 (A − A)
(2.4)
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M.I. Gil’
Similarly, taking into account that
˜ j )),
˜ T˜ = f (S)
˜ = diag (f (λ
T −1 f (A)T = f (S) = diag (f (λj )) and T˜ −1 f (A)
we get (2.3), as claimed.
¿From the previous lemma we obtain at once:
Corollary 2.2. Under the hypotheses of Theorem 1.1, we have
˜ ≤ κT κ
N2 (f (A) − f (A))
˜T [
n
X
j,k=1
˜ j )|2 ]1/2 ,
|f (λk ) − f (λ
where
κT := kT kkT −1k and κ
˜T := kT˜kkT˜ −1k.
Recall that
˜j )
f (λk ) − f (λ
˜j .
= 0 if λk = λ
˜j
λk − λ
Note that (2.3) implies
˜ ≤ kT˜ −1 k2 kT k2
N22 (f (A) − f (A))
n
X
˜j )
f (λk ) − f (λ
˜j )(T −1 T˜dj , dk )|2
|
(λk − λ
˜j
λk − λ
j,k=1
˜ j ) 2
f (λk ) − f (λ
J.
≤ kT˜ −1 k2 kT k2 max
˜j
j,k
λk − λ
(2.5)
It follows from (2.4) that
˜
J = N22 (T −1 AT˜ − T −1 A˜T˜) ≤ kT −1 k2 kT˜k2 N22 (A − A).
Thus, (2.5) yields:
Corollary 2.3. Under the hypotheses of Theorem 1.1, the inequality
is true.
˜
˜ ≤ κT κ
˜ max f (λk ) − f (λj )
N2 (f (A) − f (A))
˜ T N2 (A − A)
˜j
j,k
λk − λ
Since
˜j )
f (λk ) − f (λ
≤ sup |f ′ (s)|,
˜
˜
λk − λj
s∈co(A,A)
(2.6)
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Perturbations of Functions of Diagonalizable Matrices
˜ is the closed convex hull of the eigenvalues of the both matrices A and
where co(A, A)
˜
A, cf. [17], we have:
Corollary 2.4. Let A and A˜ be n × n diagonalizable matrices and f be a
˜ Then
function defined and differentiable on co(A, A).
˜ ≤ κT κ
N2 (f (A) − f (A))
˜T
max
˜
s∈co(A,A)
˜
|f ′ (s)|N2 (A − A).
We need also the following corollary of Lemma 2.1.
Corollary 2.5. Under the hypotheses of Theorem 1.1, it holds that
˜ j )|[
˜ ≤ kT˜ −1 kkT k max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
n
X
j=1
kT˜dj k2
n
X
k=1
k(T −1 )∗ dk k2 ]1/2 .
3. Proof of Theorem 1.1. Let Qj be the Riesz projection for µj :
Z
1
Qj = −
Rλ (A)dλ,
2πi Lj
(2.7)
(3.1)
where Lj := {z ∈ C : |z − µj | = δj }.
Lemma 3.1. The inequality
kQj k ≤
n−1
X
k=0
g k (A)
√
δjk k!
is true.
Proof. By the Schur theorem, there is an orthonormal basis {ek }, in which A =
D + V , where D = diag (λj ) is a normal matrix (the diagonal part of A) and V
is an upper triangular nilpotent matrix (the nilpotent part of A). Let Pj be the
eigenprojection of D corresponding to µj . Then by (3.1),
Z
Z
1
1
Qj − Pj = −
(Rλ (A) − Rλ (D))dλ =
Rλ (A)V Rλ (D)dλ
2πi Lj
2πi Lj
since V = A − D. But
Rλ (A) = (D + V − Iλ)−1 = (I + Rλ (D)V )−1 Rλ (D).
Consequently,
Rλ (A) =
n−1
X
(−1)k (Rλ (D)V )k Rλ (D).
k=0
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M.I. Gil’
So, we have
Qj − Pj =
n−1
X
Ck ,
(3.2)
k=1
where
Ck = (−1)k
1
2πi
Z
(Rλ (D)V )k Rλ (D)dλ.
Lj
By Theorem 2.5.1 of [8], we get
k(Rλ (D)V )k k ≤
N2k (V )
N2k (Rλ (D)V )
√
√ ,
≤
k!
ρk (A, λ) k!
where ρ(A, λ) = mink |λ − λk |. In addition, directly from the definition of g(A), when
the Hilbert-Schmidt norm is calculated in the basis {ek }, we have N2 (V ) = g(A). So,
k(Rλ (D)V )k k ≤
g k (A)
√
(λ ∈ Lj ),
δjk k!
and therefore,
kCk k ≤
1
2π
Z
Lj
k(Rλ (D)V )k Rλ (D)k|dλ| ≤
g k (A) 1
√
k+1
δj
k! 2π
Z
Lj
|dλ| =
g k (A)
√ .
δjk k!
Hence,
kQj − Pj k ≤
n−1
X
k=1
kCk k ≤
n−1
X
k=1
g k (A)
√ ,
δjk k!
(3.3)
and therefore,
kQj k ≤ kPj k + kQj − Pj k ≤
n−1
X
k=0
g k (A)
√ ,
δjk k!
as claimed.
Let {vk }nk=1 be a sequence of the eigenvectors of A, and {uk }nk=1 be the biorthogonal sequence: (vj , uk ) = 0 (j 6= k), (vj , uj ) = 1 (j, k = 1, . . . , n). So
A=
m
X
µk Q k =
k=1
n
X
λk (·, uk )vk .
k=1
Rearranging these biorthogonal sequences, we can write
Qj =
pj
X
k=1
(·, ujk )vjk ,
(3.4)
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309
p
p
j
j
are biorthogonal: (vjk , ujm ) = 0 (m 6= k), (vjk , ujk ) =
and {vjk }k=1
where {ujk }k=1
1 (m, k = 1, . . . , pj ). Observe that we can always choose these systems so that kujk k =
kvjk k.
Lemma 3.2. Let kujk k = kvjk k. Then we have
kujk k2 ≤ kQj k (k = 1, . . . , pj ).
Proof. Clearly, Qj ujk = (Qj ujk , ujk )vjk . So,
(Qj ujk , vjk ) = (ujk , ujk )(vjk , vjk ) = kujk k4 .
Hence, kujk k4 ≤ kQj kkujk k2 , as claimed.
Lemmas 3.1 and 3.2 imply:
Corollary 3.3. The inequalities
kujs k2 ≤
n−1
X
k=0
g k (A)
√
(s = 1, 2, . . . , pj )
δjk k!
are valid.
Again, let {dk } be the standard orthonormal basis.
Lemma 3.4. Let A be an n × n diagonalizable matrix. Then the operator
T =
n
X
(·, dk )vk
(3.5)
k=1
has the inverse one defined by
T −1 =
n
X
(·, uk )dk ,
(3.6)
k=1
and (2.1) holds.
Proof. Indeed, we can write out
T −1 T =
n
X
j=1
dj
n
X
(·, dk )(vk , uj ) =
n
X
dj (·, dj ) = I
j=1
k=1
and
T T −1 =
n
X
k=1
(·, uk )
n
X
j=1
(dk , dj )vj =
n
X
k=1
(·, uk )vk = I.
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M.I. Gil’
Moreover,
AT =
n
X
vj λj (·, dj ) and S =
j=1
n
X
λk (·, dk )dk .
k=1
Hence,
T
−1
AT =
n
X
dk
n
X
(vj , uk )λj (·, dj ) =
dj λj (·, dj ) = S.
j=1
j=1
k=1
n
X
So, (2.1) really holds.
Lemma 3.5. Let T be defined by (3.5). Then
kT k2 ≤
m
X
j=1
pj kQj k and kT −1 k2 ≤
m
X
j=1
pj kQj k,
and therefore,
κT ≤
m
X
j=1
pj kQj k.
Proof. Due to the previous lemma
kT −1 xk2 =
n
X
k=1
|(x, uk )|2 ≤ kxk2
n
X
k=1
kuk k2 (x ∈ Cn ).
(3.7)
By the Schwarz inequality,
(T x, T x) =
n
n X
X
(x, dk )(vk , vs )(x, ds )
k=1 s=1
≤
n
X
k=1
|(x, dk )|2 [
n
X
s,k=1
|(vk , vs )|2 ]1/2 ≤ kxk2
n
X
s=1
kvs k2 .
But by Lemma 3.2,
n
X
s=1
kus k2 =
n
X
s=1
kvs k2 =
pj
m X
X
j=1 k=1
Now (3.7) and (3.8) yield the required result.
Lemmas 3.1 and 3.5 imply:
kvjk k2 ≤
m
X
j=1
pj kQj k.
(3.8)
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Perturbations of Functions of Diagonalizable Matrices
Corollary 3.6. The inequality κT ≤ β(A) is true.
Lemma 3.7. Under the hypotheses of Theorem 1.1, we have
˜ j )|[
˜ ≤ kT˜ −1 kkT k max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
m
˜
X
j=1
˜jk
p˜j kQ
m
X
k=1
pk kQk k]1/2 ,
˜ j are the eigenprojections of A and A,
˜ respectively.
where Qk , Q
Proof. By (3.6), we have
(T −1 )∗ =
n
X
(·, dk )uk .
k=1
Moreover, due to Lemma 3.2,
n
X
j=1
k(T −1 )∗ dj k2 =
n
X
k=1
kuk k2 ≤
m
X
k=1
pk kQk k.
Similarly,
n
X
k=1
kT˜dk k2 ≤
m
X
k=1
˜ k k.
pk k Q
Now (2.7) implies the required result.
Thanks to Lemmas 3.5 and 3.7 we get the inequality
˜ j )|
˜ ≤ max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
m
˜
X
j=1
˜jk
p˜j kQ
m
X
k=1
pk kQk k.
(3.9)
Proof of Theorem 1.1: Inequality (1.2) is due to (2.6) and Corollary 3.6. Inequality (1.3) is due to (3.9) and Lemma 3.1.
4. Applications. Let us consider the two differential equations
du
= Au (t > 0) and
dt
d˜
u
˜u (t > 0),
= A˜
dt
whose solutions are u(t) and u
˜(t), respectively. Let us take the initial conditions
u(0) = u˜(0) ∈ Cn . Then we have
˜
u(t) − u˜(t) = (exp [At] − exp [At])u(0).
By (1.2),
˜
ku(t) − u˜(t)k ≤ ku(0)kN2(exp [At] − exp [At])
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M.I. Gil’
˜
˜
≤ tetα N2 (A − A)β(A)β(
A)ku(0)k
(t ≥ 0),
where
˜j }.
α = max{max Re λk , max Re λ
k
j
Furthermore, let 0 6∈ σ(A) ∪ σ(A). Then the function
A−1/2 sin (A1/2 t) (t > 0)
is the Green function to the Cauchy problem for the equation
d2 w
+ Aw = 0 (t > 0).
dt2
Simultaneously, consider the equation
d2 w
˜
+ A˜w
˜ = 0 (t > 0),
dt2
and take the initial conditions
w(0) = w(0)
˜
∈ Cn and w′ (0) = w
˜ ′ (0) = 0.
Then we have
w(t) − w(t)
˜ = [A−1/2 sin (A1/2 t) − A˜−1/2 sin (A˜1/2 t)]w(0).
By (1.2),
√
sin (t√λk )
˜j )
sin (t λ
√
√
−
λk
˜j
λ
˜
˜
kw(t) − w(t)k
˜
≤ β(A)β(A)N2 (A − A) max
˜
j,k
λk − λj
(t > 0).
Here one can take an arbitrary branch of the roots.
Acknowledgment. I am very grateful to the referee of this paper for his really
helpful remarks.
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Perturbations of Functions of Diagonalizable Matrices
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[19] L. Verde-Star. Functions of matrices. Linear Algebra Appl., 406:285–300, 2005.
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PERTURBATIONS OF FUNCTIONS OF
DIAGONALIZABLE MATRICES∗
MICHAEL I. GIL’†
˜ be n × n diagonalizable matrices and f be a function defined on their
Abstract. Let A and A
˜ are established. Applications to
spectra. In the present paper, bounds for the norm of f (A) − f (A)
differential equations are also discussed.
Key words. Matrix valued functions, Perturbations, Similarity of matrices, Diagonalizable
matrices.
AMS subject classifications. 15A54, 15A45, 15A60.
n
1. Introduction and statement of the main result.
p Let C be a Euclidean
space with the scalar product (·, ·), Euclidean norm k·k = (·, ·) and identity operator
˜j (j = 1, . . . , n), respectively.
I. A and A˜ are n × n matrices with eigenvalues λj and λ
σ(A) denotes the spectrum of A, A∗ is the adjoint to A, and N2 (A) is the HilbertSchmidt (Frobenius) norm of A: N22 (A) = Trace(A∗ A).
In the sequel, it is assumed that each of the matrices A and A˜ has n linearly
independent eigenvectors, and therefore, these matrices are diagonalizable. In other
words, the eigenvalues of these matrices are semi-simple.
˜ The aim of this paper is to
Let f be a scalar function defined on σ(A) ∪ σ(A).
˜
establish inequalities for the norm of f (A) − f (A). The literature on perturbations
of matrix valued functions is very rich but mainly, perturbations of matrix functions
of a complex argument and matrix functions of Hermitian matrices were considered,
cf. [1, 11, 13, 14, 16, 18]. The matrix valued functions of a non-Hermitian argument
have been investigated essentially less, although they are very important for various
applications; see the book [10].
The following quantity plays an essential role hereafter:
g(A) :=
"
N22 (A)
−
n
X
k=1
|λk |
2
#1/2
.
∗ Received by the editors December 8, 2009. Accepted for publication April 19, 2010. Handling
Editor: Peter Lancaster.
† Department of Mathematics, Ben Gurion University of the Negev, P.0. Box 653, Beer-Sheva
84105, Israel ([email protected]). Supported by the Kameah fund of the Israel.
303
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M.I. Gil’
g(A) enjoys the following properties:
g 2 (A) ≤ 2N22 (AI ) (AI = (A − A∗ )/2i) and g 2 (A) ≤ N22 (A) − |Trace(A2 )|, (1.1)
cf. [8, Section 2.1]. If A is normal, then g(A) = 0. Denote by µj , j = 1, . . . , m ≤ n,
the distinct eigenvalues of A, and by pj the algebraic multiplicity of µj . In particular,
˜j ,
one can write µ1 = λ1 = · · · = λp1 , µ2 = λp1 +1 = · · · = λp1 +p2 , etc. Similarly, µ
˜ and p˜j denotes the algebraic
j = 1, . . . , m
˜ ≤ n, are the distinct eigenvalues of A,
multiplicity of µ
˜j .
Let δj be the half-distance from µj to the other eigenvalues of A, namely,
δj :=
min
|µj − µk |/2 > 0.
min
|˜
µj − µ
˜k |/2 > 0.
k=1,...,m; k6=j
Similarly,
δ˜j :=
k=1,...,m;
˜ k6=j
Put
β(A) :=
m
X
j=1
pj
n−1
X
k=0
g k (A)
√
δjk k!
˜ :=
and β(A)
m
˜
X
p˜j
j=1
n−1
X
k=0
˜
g k (A)
√ .
δ˜k k!
j
Here g 0 (A) = δj0 = 1 and g˜0 (A) = δ˜j0 = 1. According to (1.1),
√
n−1
m
X ( 2N2 (AI ))k
X
√
pj
β(A) ≤
.
δjk k!
j=1
k=0
In what follows, we put
˜j )
f (λk ) − f (λ
˜j .
= 0 if λk = λ
˜j
λk − λ
Now we are in a position to formulate our main result.
Theorem 1.1. Let A and A˜ be n × n diagonalizable matrices and f be a function
˜ Then the inequalities
defined on σ(A) ∪ σ(A).
f (λ ) − f (λ
˜ j )
k
˜
˜
˜
N2 (f (A) − f (A)) ≤ β(A)β(A) max
(1.2)
N2 (A − A)
˜j
j,k
λk − λ
and
˜ j )|
˜ ≤ β(A)β(A)
˜ max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
(1.3)
are valid.
The proof of this theorem is divided into lemmas which are presented in the next
two sections. The importance of Theorem 1.1 lies in the fact that the right-hand sides
˜
of inequalities (1.2) and (1.3) only involve universal quantities calculated for A and A,
˜
and the values of the function f on the spectra σ(A) and σ(A), but e.g. no matrices
performing similarities of A and A˜ to diagonal ones.
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Perturbations of Functions of Diagonalizable Matrices
305
2. The basic lemma. Let T and T˜ be the invertible matrices performing similarities of A and A˜ to diagonal ones, that is,
T −1 AT = S,
(2.1)
˜
T˜ −1 A˜T˜ = S,
(2.2)
and
˜ j ). Everywhere below, {dk }n is the standard
where S = diag (λj ) and S˜ = diag (λ
k=1
orthonormal basis and kBk = sup {kBhk/khk : h ∈ Cn } for an n × n matrix B.
Recall that N2 (AB) ≤ kBkN2 (A).
We begin with the following lemma.
Lemma 2.1. Let the hypotheses of Theorem 1.1 hold. Then
˜ ≤ kT˜ −1k2 kT k2
N22 (f (A) − f (A))
n
X
j,k=1
˜ j ))(T −1 T˜dj , dk )|2 .
|(f (λk ) − f (λ
(2.3)
Proof. We have
˜ = N 2 (T ST −1 − T˜S˜T˜ −1 ) = N 2 (T ST −1T˜T˜ −1 − T T −1T˜S˜T˜ −1 )
N22 (A − A)
2
2
˜ T˜ −1 ) ≤ kT˜ −1 k2 kT k2N 2 (ST −1 T˜ − T −1 T˜S).
˜
= N22 (T (ST −1T˜ − T −1 T˜S)
2
But
˜ =
N22 (ST −1 T˜ − T −1 T˜S)
=
n
X
j,k=1
n
X
j,k=1
˜ j , dk )|2
|(ST −1 T˜ − T −1 T˜S)d
˜ j , dk )|2 = J,
|(T −1 T˜dj , S ∗ dk ) − (T −1 T˜Sd
where
J :=
n
X
j,k=1
˜j )(T −1 T˜dj , dk )|2 .
|(λk − λ
So,
˜ ≤ kT˜ −1k2 kT k2 J.
N22 (A − A)
(2.4)
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Similarly, taking into account that
˜ j )),
˜ T˜ = f (S)
˜ = diag (f (λ
T −1 f (A)T = f (S) = diag (f (λj )) and T˜ −1 f (A)
we get (2.3), as claimed.
¿From the previous lemma we obtain at once:
Corollary 2.2. Under the hypotheses of Theorem 1.1, we have
˜ ≤ κT κ
N2 (f (A) − f (A))
˜T [
n
X
j,k=1
˜ j )|2 ]1/2 ,
|f (λk ) − f (λ
where
κT := kT kkT −1k and κ
˜T := kT˜kkT˜ −1k.
Recall that
˜j )
f (λk ) − f (λ
˜j .
= 0 if λk = λ
˜j
λk − λ
Note that (2.3) implies
˜ ≤ kT˜ −1 k2 kT k2
N22 (f (A) − f (A))
n
X
˜j )
f (λk ) − f (λ
˜j )(T −1 T˜dj , dk )|2
|
(λk − λ
˜j
λk − λ
j,k=1
˜ j ) 2
f (λk ) − f (λ
J.
≤ kT˜ −1 k2 kT k2 max
˜j
j,k
λk − λ
(2.5)
It follows from (2.4) that
˜
J = N22 (T −1 AT˜ − T −1 A˜T˜) ≤ kT −1 k2 kT˜k2 N22 (A − A).
Thus, (2.5) yields:
Corollary 2.3. Under the hypotheses of Theorem 1.1, the inequality
is true.
˜
˜ ≤ κT κ
˜ max f (λk ) − f (λj )
N2 (f (A) − f (A))
˜ T N2 (A − A)
˜j
j,k
λk − λ
Since
˜j )
f (λk ) − f (λ
≤ sup |f ′ (s)|,
˜
˜
λk − λj
s∈co(A,A)
(2.6)
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Perturbations of Functions of Diagonalizable Matrices
˜ is the closed convex hull of the eigenvalues of the both matrices A and
where co(A, A)
˜
A, cf. [17], we have:
Corollary 2.4. Let A and A˜ be n × n diagonalizable matrices and f be a
˜ Then
function defined and differentiable on co(A, A).
˜ ≤ κT κ
N2 (f (A) − f (A))
˜T
max
˜
s∈co(A,A)
˜
|f ′ (s)|N2 (A − A).
We need also the following corollary of Lemma 2.1.
Corollary 2.5. Under the hypotheses of Theorem 1.1, it holds that
˜ j )|[
˜ ≤ kT˜ −1 kkT k max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
n
X
j=1
kT˜dj k2
n
X
k=1
k(T −1 )∗ dk k2 ]1/2 .
3. Proof of Theorem 1.1. Let Qj be the Riesz projection for µj :
Z
1
Qj = −
Rλ (A)dλ,
2πi Lj
(2.7)
(3.1)
where Lj := {z ∈ C : |z − µj | = δj }.
Lemma 3.1. The inequality
kQj k ≤
n−1
X
k=0
g k (A)
√
δjk k!
is true.
Proof. By the Schur theorem, there is an orthonormal basis {ek }, in which A =
D + V , where D = diag (λj ) is a normal matrix (the diagonal part of A) and V
is an upper triangular nilpotent matrix (the nilpotent part of A). Let Pj be the
eigenprojection of D corresponding to µj . Then by (3.1),
Z
Z
1
1
Qj − Pj = −
(Rλ (A) − Rλ (D))dλ =
Rλ (A)V Rλ (D)dλ
2πi Lj
2πi Lj
since V = A − D. But
Rλ (A) = (D + V − Iλ)−1 = (I + Rλ (D)V )−1 Rλ (D).
Consequently,
Rλ (A) =
n−1
X
(−1)k (Rλ (D)V )k Rλ (D).
k=0
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So, we have
Qj − Pj =
n−1
X
Ck ,
(3.2)
k=1
where
Ck = (−1)k
1
2πi
Z
(Rλ (D)V )k Rλ (D)dλ.
Lj
By Theorem 2.5.1 of [8], we get
k(Rλ (D)V )k k ≤
N2k (V )
N2k (Rλ (D)V )
√
√ ,
≤
k!
ρk (A, λ) k!
where ρ(A, λ) = mink |λ − λk |. In addition, directly from the definition of g(A), when
the Hilbert-Schmidt norm is calculated in the basis {ek }, we have N2 (V ) = g(A). So,
k(Rλ (D)V )k k ≤
g k (A)
√
(λ ∈ Lj ),
δjk k!
and therefore,
kCk k ≤
1
2π
Z
Lj
k(Rλ (D)V )k Rλ (D)k|dλ| ≤
g k (A) 1
√
k+1
δj
k! 2π
Z
Lj
|dλ| =
g k (A)
√ .
δjk k!
Hence,
kQj − Pj k ≤
n−1
X
k=1
kCk k ≤
n−1
X
k=1
g k (A)
√ ,
δjk k!
(3.3)
and therefore,
kQj k ≤ kPj k + kQj − Pj k ≤
n−1
X
k=0
g k (A)
√ ,
δjk k!
as claimed.
Let {vk }nk=1 be a sequence of the eigenvectors of A, and {uk }nk=1 be the biorthogonal sequence: (vj , uk ) = 0 (j 6= k), (vj , uj ) = 1 (j, k = 1, . . . , n). So
A=
m
X
µk Q k =
k=1
n
X
λk (·, uk )vk .
k=1
Rearranging these biorthogonal sequences, we can write
Qj =
pj
X
k=1
(·, ujk )vjk ,
(3.4)
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309
p
p
j
j
are biorthogonal: (vjk , ujm ) = 0 (m 6= k), (vjk , ujk ) =
and {vjk }k=1
where {ujk }k=1
1 (m, k = 1, . . . , pj ). Observe that we can always choose these systems so that kujk k =
kvjk k.
Lemma 3.2. Let kujk k = kvjk k. Then we have
kujk k2 ≤ kQj k (k = 1, . . . , pj ).
Proof. Clearly, Qj ujk = (Qj ujk , ujk )vjk . So,
(Qj ujk , vjk ) = (ujk , ujk )(vjk , vjk ) = kujk k4 .
Hence, kujk k4 ≤ kQj kkujk k2 , as claimed.
Lemmas 3.1 and 3.2 imply:
Corollary 3.3. The inequalities
kujs k2 ≤
n−1
X
k=0
g k (A)
√
(s = 1, 2, . . . , pj )
δjk k!
are valid.
Again, let {dk } be the standard orthonormal basis.
Lemma 3.4. Let A be an n × n diagonalizable matrix. Then the operator
T =
n
X
(·, dk )vk
(3.5)
k=1
has the inverse one defined by
T −1 =
n
X
(·, uk )dk ,
(3.6)
k=1
and (2.1) holds.
Proof. Indeed, we can write out
T −1 T =
n
X
j=1
dj
n
X
(·, dk )(vk , uj ) =
n
X
dj (·, dj ) = I
j=1
k=1
and
T T −1 =
n
X
k=1
(·, uk )
n
X
j=1
(dk , dj )vj =
n
X
k=1
(·, uk )vk = I.
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M.I. Gil’
Moreover,
AT =
n
X
vj λj (·, dj ) and S =
j=1
n
X
λk (·, dk )dk .
k=1
Hence,
T
−1
AT =
n
X
dk
n
X
(vj , uk )λj (·, dj ) =
dj λj (·, dj ) = S.
j=1
j=1
k=1
n
X
So, (2.1) really holds.
Lemma 3.5. Let T be defined by (3.5). Then
kT k2 ≤
m
X
j=1
pj kQj k and kT −1 k2 ≤
m
X
j=1
pj kQj k,
and therefore,
κT ≤
m
X
j=1
pj kQj k.
Proof. Due to the previous lemma
kT −1 xk2 =
n
X
k=1
|(x, uk )|2 ≤ kxk2
n
X
k=1
kuk k2 (x ∈ Cn ).
(3.7)
By the Schwarz inequality,
(T x, T x) =
n
n X
X
(x, dk )(vk , vs )(x, ds )
k=1 s=1
≤
n
X
k=1
|(x, dk )|2 [
n
X
s,k=1
|(vk , vs )|2 ]1/2 ≤ kxk2
n
X
s=1
kvs k2 .
But by Lemma 3.2,
n
X
s=1
kus k2 =
n
X
s=1
kvs k2 =
pj
m X
X
j=1 k=1
Now (3.7) and (3.8) yield the required result.
Lemmas 3.1 and 3.5 imply:
kvjk k2 ≤
m
X
j=1
pj kQj k.
(3.8)
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Perturbations of Functions of Diagonalizable Matrices
Corollary 3.6. The inequality κT ≤ β(A) is true.
Lemma 3.7. Under the hypotheses of Theorem 1.1, we have
˜ j )|[
˜ ≤ kT˜ −1 kkT k max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
m
˜
X
j=1
˜jk
p˜j kQ
m
X
k=1
pk kQk k]1/2 ,
˜ j are the eigenprojections of A and A,
˜ respectively.
where Qk , Q
Proof. By (3.6), we have
(T −1 )∗ =
n
X
(·, dk )uk .
k=1
Moreover, due to Lemma 3.2,
n
X
j=1
k(T −1 )∗ dj k2 =
n
X
k=1
kuk k2 ≤
m
X
k=1
pk kQk k.
Similarly,
n
X
k=1
kT˜dk k2 ≤
m
X
k=1
˜ k k.
pk k Q
Now (2.7) implies the required result.
Thanks to Lemmas 3.5 and 3.7 we get the inequality
˜ j )|
˜ ≤ max |f (λk ) − f (λ
N2 (f (A) − f (A))
j,k
m
˜
X
j=1
˜jk
p˜j kQ
m
X
k=1
pk kQk k.
(3.9)
Proof of Theorem 1.1: Inequality (1.2) is due to (2.6) and Corollary 3.6. Inequality (1.3) is due to (3.9) and Lemma 3.1.
4. Applications. Let us consider the two differential equations
du
= Au (t > 0) and
dt
d˜
u
˜u (t > 0),
= A˜
dt
whose solutions are u(t) and u
˜(t), respectively. Let us take the initial conditions
u(0) = u˜(0) ∈ Cn . Then we have
˜
u(t) − u˜(t) = (exp [At] − exp [At])u(0).
By (1.2),
˜
ku(t) − u˜(t)k ≤ ku(0)kN2(exp [At] − exp [At])
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M.I. Gil’
˜
˜
≤ tetα N2 (A − A)β(A)β(
A)ku(0)k
(t ≥ 0),
where
˜j }.
α = max{max Re λk , max Re λ
k
j
Furthermore, let 0 6∈ σ(A) ∪ σ(A). Then the function
A−1/2 sin (A1/2 t) (t > 0)
is the Green function to the Cauchy problem for the equation
d2 w
+ Aw = 0 (t > 0).
dt2
Simultaneously, consider the equation
d2 w
˜
+ A˜w
˜ = 0 (t > 0),
dt2
and take the initial conditions
w(0) = w(0)
˜
∈ Cn and w′ (0) = w
˜ ′ (0) = 0.
Then we have
w(t) − w(t)
˜ = [A−1/2 sin (A1/2 t) − A˜−1/2 sin (A˜1/2 t)]w(0).
By (1.2),
√
sin (t√λk )
˜j )
sin (t λ
√
√
−
λk
˜j
λ
˜
˜
kw(t) − w(t)k
˜
≤ β(A)β(A)N2 (A − A) max
˜
j,k
λk − λj
(t > 0).
Here one can take an arbitrary branch of the roots.
Acknowledgment. I am very grateful to the referee of this paper for his really
helpful remarks.
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