R E S E A R C H
Open Access
Perturbation bounds for eigenvalues of
diagonalizable matrices and singular values
Duanmei Zhou
1*, Guoliang Chen
2, Guoxing Wu
3and Xiaoyan Chen
4*Correspondence:
[email protected] 1College of Mathematics and
Computer Science, Gannan Normal University, 341000 Ganzhou, People’s Republic of China Full list of author information is available at the end of the article
Abstract
Perturbation bounds for eigenvalues of diagonalizable matrices are derived. Perturbation bounds for singular values of arbitrary matrices are also given. We generalize some existing results.
MSC: 65F15; 15A18; 65F35; 15A42
Keywords: perturbation bound; eigenvalue; diagonalizable matrix; singular value
1 Introduction
Many problems in science and engineering lead to eigenvalue and singular value problems for matrices. Perturbation bounds of eigenvalues and singular values play an important role in matrix computations. LetSnbe the set of alln! permutations of{, , . . . ,n}. Ifx=
(x,x, . . . ,xn) andπ∈Sn, then the vectorxπ is defined as (xπ(),xπ(), . . . ,xπ(n)). A square matrix is calleddoubly stochasticif its elements are real nonnegative numbers and if the sum of the elements in each row and in each column is equal to . LetCn×nbe the set of
n×ncomplex matrices. LetA= (aij)∈Cn×n, we use the notation (see [, ])
Ap=
n
i,j=
|aij|p
p
forp≥, (.)
Aq,p=
n
j=
n
k=
|akj|q
p qp
forp> ,q> , p+
q = . (.)
LetT∈Cn×nand assume that
(k)=diagλ(k),λ(k), . . . ,λ(nk)∈Cn×n, k= , . . . , ,
are diagonal matrices. In [], the following classical result is given:
()T()–()T()≥sn(T)
n
i=
λ()i λ()π(i)–λ()i λ()π(i) (.)
for someπ∈Sn, wheresn(T) is the smallest singular value ofT. The inequality has many
applications in bounding the (relative) perturbation for eigenvalues and singular values, such as [–] and the references therein. We generalize (.) in Section .
Letλ(A) denote the spectrum of matrixA. In , Ikramov [] defined the ‘Hölder dis-tancedp(λ(A),λ(B)) between the spectra’ of the matricesAandB, which have the
eigen-valuesλ,λ, . . . ,λnandμ,μ, . . . ,μn, respectively, by the equation:
dp
λ(A),λ(B)=min
π∈Sn
n
i=
|λi–μπ(i)|p
p
. (.)
IfAandBare Hermitian matrices and ≤p< , [] obtained
dp
λ(A),λ(B)≤ A–Bp, (.)
which partially generalizes the Hoffman-Wielandt theorem []. However, for normal ma-trices (.) can no longer be valid. The purpose of this paper is to obtain several inequali-ties similar to (.) for diagonalizable matrices. We exhibit some upper bounds and lower bounds fordp(λ(A),λ(B)) of diagonalizable matricesAandBin Section .
Majorization is one of the most powerful techniques for deriving inequalities. We use majorization to get some perturbation bounds for singular values. For simplicity of the notations, in most cases in this paper the vectors inRnare regarded as row vectors, but
when they are multiplied by matrices we regard them as column vectors. Given a real vectorx= (x,x, . . . ,xn)∈Rn, we rearrange its components asx[]≥x[]≥ · · · ≥x[n]. Definition .([], p.) Forx= (x,x, . . . ,xn),y= (y,y, . . . ,yn)∈Rn, if
k
i= x[i]≤
k
i=
y[i], k= , , . . . ,n,
then we say thatxis weakly majorized byyand denotex≺wy. If x≺wyand ni=xi= n
i=yi, then we say thatxis majorized byyand denotex≺y.
Lets≥s≥ · · · ≥snandδ≥δ≥ · · · ≥δnbe the singular values of the complex
ma-tricesA= (aij)∈Cn×nandB= (bij)∈Cn×n, respectively. In [], p., and [], p., the
following classical result is given:
n
i=
|si–δi|≤ n
i,j=
|aij–bij|. (.)
We generalize the inequality (.) in Section .
2 Perturbation bounds for eigenvalues of diagonalizable matrices
LetA◦Bdenote the Hadamard product of matricesAandB.Adenotes the spectral norm of matrixA.ATdenotes the transpose of matrixA. For twon-square real matricesA,
B, we writeA≤eBto mean thatB–Ais (entrywise) nonnegative. ForA= (aij)∈Cn×nand a
real numbert> , we denoteA|◦|t≡(|a
the largest singular values ofA, respectively. The following entrywise inequalities involve the smallest and the largest singular values.
Lemma .([], p.) Let A∈Cn×nand let p,q be real numbers with <p≤and q≥.
Then there exist two doubly stochastic matrices B,C∈Cn×nsuch that
sn(A)pB≤eA|◦|p (.)
and
A|◦|q≤es(A)qC. (.)
Theorem . Let T∈Cn×nand let p,q be real numbers with <p≤and q≥.Assume that(k)=diag(λ(k),λ(k), . . . ,λ(nk))∈Cn×n,k= , . . . , ,are diagonal matrices.Then there are
permutationsπandνof Snsuch that
()T()–()T()pp≥spn(T)
n
i=
λ()i λ()π(i)–λ()i λ()π(i)p (.)
and
()T()–()T()q
q≤s q
(T)
n
i=
λ()i λ()ν(i)–λ()i λ()ν(i)q. (.)
Proof SetT= (tij)∈Cn×n. Then
()T()–()T()p
p= n
i,j=
|tij|pλ()i λ
()
j –λ
()
i λ
()
j p
=eTT|◦|p◦Me, (.)
whereM= (|λ()i λ()j –λ()i λj()|p)∈Cn×n,e= (, , . . . , )T∈Cn. Applying inequality (.), we
have
T|◦|p≥spn(T)B,
whereB= (bij) is a doubly stochastic matrix. Then
()T()–()T()pp≥eTspn(T)B◦Me=spn(T)eT(B◦M)e.
SinceBis doubly stochastic, by Birkhoff ’s theorem ([, ], p.)Bis a convex combi-nation of permutation matrices:
B=
n!
i=
τiPi, τi≥, n!
i=
SupposeeT(B◦P
k)e=min{eT(B◦Pi)e|≤i≤n!}andPkcorresponds toπ∈Sn. Then
()T()–()T()pp≥spn(T)eT(B◦M)e
=spn(T)eT
n!
i=
τiPi◦M
e
≥spn(T)
n!
i=
τieT(Pk◦M)e
=spn(T)eT(Pk◦M)e
=spn(T)
n
i=
λ()i λ()π(i)–λ()i λ()π(i)p.
Proving (.).
Use (.), (.), and the Birkhoff theorem, we can deduce the inequality (.).
Remark . If we takep= , we get Theorem . in []. So, the bound in inequality (.)
generalizes the bound of Theorem . in [].
Next, we apply Theorem . to get some perturbation bounds for the eigenvalues of diagonalizable matrices. LetA= (aij)∈Cn×nandB= (bij)∈Cn×n. Whenp> ,q> , p+
q= , then (see [])
ABp≤ ApBq,p, (.)
ABp≤ATq,pBp. (.)
IfBis nonsingular, then we have
Ap
B–
q,p
≤ ABp. (.)
IfAis nonsingular, then we have
Bp
(A–)T q,p
≤ ABp. (.)
For normal matrices the statement of Theorem in [] (inequality (.)) can no longer be valid. However, we have the following theorem.
Theorem . Assume that both A∈Cn×nand B∈Cn×nare diagonalizable and admit the following decompositions:
where D and D are nonsingular,and=diag(λ,λ, . . . ,λn)and=diag(μ,μ, . . . , μn).Then there are permutationsπandνof Snsuch that
n
i=
|λi–μπ(i)|p
p
≤D– Tq,pDq,pD– DA–Bp, (.)
n
i=
|λi–μν(i)|p
p
≤D– Tq,pDq,pD– DA–Bp, (.)
where <p≤andp+q= . Proof Using (.), we have
A–Bpp=DD– –DD–
p p
=D
D– D–D– D
D– pp (.)
and
A–Bpp=DD– –DD–
p p
=D
D– D–D– D
D– pp. (.)
We give a proof of (.) with the help of (.). Similarly one can prove (.) using (.). Applying (.) and (.) to (.) we obtain
A–Bpp≥D –
D–D– Dpp
(D– )T
p
q,pDpq,p
.
Using inequality (.), there is a permutationπofSnsuch that
D– D–D–Dpp≥spn
D– D
n
i=
|λi–μπ(i)|p.
So we have
n
i=
|λi–μπ(i)|p≤
(D– )Tpq,pDpq,p
spn(D– D)
A–Bpp.
We use the relations
s–n D– D
=D– D
–
≤ DD–
to get the inequality in (.).
Theorem . Under the hypotheses of Theorem.,there are permutationsπandνof Sn
such that
n
i=
|λi–μπ(i)|q
q
≥ A–Bq
(D)Tp,qD–p,qD– D
n
i=
|λi–μν(i)|q
q
≥ A–Bq
(D)Tp,qD–p,qDD–
, (.)
where q≥andp+q= . Proof Using (.), we have
A–Bqq=DD– –DD–
q q
=D
D– D–D– D
D– qq (.)
and
A–Bqq=DD– –DD–
q q
=D
D– D–D– D
D– qq. (.)
Applying (.) and (.) to (.) we obtain
A–Bqq≤(D)T
q p,qD
–
D–D– D
q qD
–
q p,q.
Using inequality (.), there exists a permutationπofSnsuch that
D– D–D– D
q q≤s
q
D– D
n
i=
|λi–μπ(i)|q,
so we have
n
i=
|λi–μπ(i)|q≥ A–Bqq
(D)Tqp,qD–
q
p,qsq(D– D) .
We use the relations
s
D– D
=D– D≤D– D
to get the inequality in (.).
The proof of inequality (.) is similar to the proof of (.) and is omitted here.
For ≤p≤, it is well known [] that the scalar function (.) of a matrixAis a sub-multiplicative matrix norm. However, it is true for <p≤. Actually, according to the Cauchy-Schwarz inequality, we have
ABpp=
n
i=
n
j=
n
k= aikbkj
p
≤
n
i=
n
j=
n
k=
|aik|
n
k=
|bkj|
p
Since for fixed vectorx= (x,x, . . . ,xn), the functionp→(|x|p+|x|p+· · ·+|xn|p)
p is
decreasing on (,∞),
ABpp≤
n
i=
n
j=
n
k=
|aik|p
n
k=
|bkj|p
=
n
i=
n
k=
|aik|p
n
j=
n
k=
|bkj|p
=AppBpp. That is,
ABp≤ ApBp. (.)
IfBis nonsingular, then
Ap=ABB–p≤ ABpB–p.
So we have
Ap
B–
p ≤
ABp. (.)
Similarly, whenAis nonsingular, then
Bp=A–ABp≤A–pABp.
That is,
Bp
A–
p ≤
ABp. (.)
Theorem . Under the assumptions of Theorem.,there are permutationsπandνof Snsuch that
n
i=
|λi–μπ(i)|p
p
≤D– pDpD– DA–Bp, (.)
n
i=
|λi–μν(i)|p
p
≤D– pDpD– DA–Bp, (.)
where <p≤.
Proof The proof is similar to the proof of Theorem . and is omitted here.
Remark . Since
Aq,p=
n
j=
n
k=
|akj|q
p qp
≤
n
j=
n
k=
|akj|p
p pp
and
ATq,p≤ATp=Ap
for <p≤ and p+q= , the bounds in (.) and (.) are always sharper than those in (.) and (.), respectively.
Whenp=q= , thenAB≤min{AB,AB}. We obtain
n
i=
|λi–μπ(i)|
≤D– DD– DA–B
≤D– DD– DA–B,
n
i=
|λi–μν(i)|
≤D– DD– DA–B
≤D– DD– DA–B. Since
Ap,q=
n
j=
n
k=
|akj|p
q pq
≤
n
j=
n
k=
|akj|p
p pp
=Ap
and
ATp,q≤ATp=Ap
forq≥ andp+q= , we have the following corollary.
Corollary . Under the same conditions as in Theorem.,there are permutationsπ andνof Snsuch that
n
i=
|λi–μπ(i)|q
q
≥ A–Bq
DpD– pD– D ,
n
i=
|λi–μν(i)|q
q
≥ A–Bq
D–
pDpDD– ,
where q≥andp+q= .
Remark . Whenp=q= , we obtain
n
i=
|λi–μπ(i)|
≥ A–B
DD– D– D≥
A–B
DD– D– D ,
n
i=
|λi–μν(i)|
≥ A–B
D–
DD– D
≥ A–B
D–
3 Perturbation bounds for singular values
For brevity we only consider square matrices. The generalizations from square matrices to rectangular matrices are obvious, and usually problems on singular values of rectangular matrices can be converted to the case of square matrices by adding zero rows or zero columns.
For a Hermitian matrix G∈Cn×n, we always denoteλ(G) = (λ
(G),λ(G), . . . ,λn(G)),
whereλ(G)≥λ(G)≥ · · · ≥λn(G) are the eigenvalues ofGin decreasing order.
Lemma .(Lidskii [], Lemma . []) If G,H∈Cn×nare Hermitian matrices,then
λ(G) –λ(H)≺λ(G–H).
Lemma .([], p.) Let f(t)be a convex function,x= (x,x, . . . ,xn),y= (y,y, . . . ,yn)∈ Rn.Then
x≺y ⇒ f(x),f(x), . . . ,f(xn)
≺w
f(y),f(y), . . . ,f(yn)
.
Theorem . Letσ(A)≥σ(A)≥ · · · ≥σn(A),σ(B)≥σ(B)≥ · · · ≥σn(B)andσ(A– B)≥σ(A–B)≥ · · · ≥σn(A–B)be the singular values of the complex matrices A= (aij),
B= (bij)and A–B,respectively.Then
n
i=
σi(A) –σi(B) p
≤
n
i,j=
|aij–bij|p, (.)
n
i=
σi(A) –σi(B) q
≥
n
i=
σiq(A–B), (.)
where≤p≤, <q≤. Proof Letϕ(A)
A∗ A
,ϕ(B)
B∗ B
. Then
ϕ(A–B) =
(A–B)∗ A–B
.
ϕ(A),ϕ(B),ϕ(A–B) are three Hermitian matrices. Assume thatU∗AV=diag(σ(A),σ(A), . . . ,σn(A)),U∗BV=diag(σ(B),σ(B), . . . ,σn(B)) andU∗(A–B)V=diag(σ(A–B),σ(A– B), . . . ,σn(A–B)) are singular value decompositions withU,U,U,V,V,V unitary. Then
Q
√
V V U –U
, Q
√
V V U –U
and Q
√
V V U –U
are unitary matrices and
Q∗ϕ(A)Q=diag
σ(A),σ(A), . . . ,σn(A), –σ(A), –σ(A), . . . , –σn(A)
,
Q∗ϕ(B)Q=diag
σ(B),σ(B), . . . ,σn(B), –σ(B), –σ(B), . . . , –σn(B)
and
Q∗ϕ(A–B)Q=diag
σ(A–B),σ(A–B), . . . ,σn(A–B),
–σ(A–B), –σ(A–B), . . . , –σn(A–B)
.
By Lemma ., we have
σ(A) –σ(B),σ(A) –σ(B), . . . ,σn(A) –σn(B),
σn(B) –σn(A), . . . ,σ(B) –σ(A),σ(B) –σ(A)
≺σ(A–B),σ(A–B), . . . ,σn(A–B),
–σn(A–B), . . . , –σ(A–B), –σ(A–B)
. (.)
First consider the case ≤p≤. Since the functionf(t) =|t|p is convex on (–∞, +∞),
applying Lemma . withf(t) to the majorization (.) yields
σ(A) –σ(B)
p
,σ(A) –σ(B)
p
, . . . ,σn(A) –σn(B) p
,
σn(B) –σn(A) p
, . . . ,σ(B) –σ(A)
p
,σ(B) –σ(A)
p
≺w
σp(A–B),σp(A–B), . . . ,σnp(A–B),σnp(A–B), . . . ,σp(A–B),σp(A–B).
In particular,
σ(A) –σ(B)
p
,σ(A) –σ(B)
p
, . . . ,σn(A) –σn(B) p
≺w
σp(A–B),σp(A–B), . . . ,σnp(A–B). (.)
According to Theorem of [] or Theorem . of [], we have
n
i=
σip(A–B)≤
n
i,j=
|aij–bij|p (.)
for ≤p≤. Combining (.) and (.), we obtain (.).
When <q≤, by considering the convex functiong(t) = –|t|q, on (–∞, +∞), applying
Lemma . withg(t) to the majorization (.) yields
–σ(A) –σ(B)
q
, . . . , –σn(A) –σn(B) q
, –σn(B) –σn(A) q
, . . . , –σ(B) –σ(A)
q
≺w
–σq(A–B), . . . , –σnq(A–B), –σnq(A–B), . . . , –σ q
(A–B)
.
In particular,
–σ(A) –σ(B)q, –σ(A) –σ(B)q, . . . , –σn(A) –σn(B)q
≺w
–σq(A–B), –σq(A–B), . . . , –σnq(A–B). (.)
From (.), we get (.).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
Author details
1College of Mathematics and Computer Science, Gannan Normal University, 341000 Ganzhou, People’s Republic of China. 2Department of Mathematics, East China Normal University, 200241 Shanghai, People’s Republic of China.3Department
of Mathematics, Northeast Forestry University, 150040 Harbin, People’s Republic of China.4Library, Gannan Normal University, 341000 Ganzhou, People’s Republic of China.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (No. 11501126, No. 11471122), the Youth Natural Science Foundation of Jiangxi Province (No. 20151BAB211011), the Science Foundation of Jiangxi Provincial Department of Education (No. GJJ150979), and the Supporting the Development for Local Colleges and Universities Foundation of China - Applied Mathematics Innovative team building.
Received: 19 January 2016 Accepted: 28 March 2016 References
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