matejas j.-relative eigenvalue and singular value perturbations of

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BIT Numerical Mathematics (2008) 48: 769–781

Published online: 2 December 2008 – c Springer 2008

DOI: 10.1007/s10543-008-0200-1

RELATIVE EIGENVALUE AND SINGULAR VALUEPERTURBATIONS OF SCALED DIAGONALLY

DOMINANT MATRICES,

J. MATEJAS1 and V. HARI2

1Faculty of Economics, University of Zagreb, Kennedyjev trg 6, 10000 Zagreb, Croatia.

email: [email protected]

2Department of Mathematics, University of Zagreb, P.O. Box 335, 10002 Zagreb, Croatia.

email: [email protected]

Abstract.

The paper derives improved relative perturbation bounds for the eigenvalues of scaled

diagonally dominant Hermitian matrices and new relative perturbation bounds for

the singular values of symmetrically scaled diagonally dominant square matrices. The

perturbation result for the singular values enlarges the class of well-behaved matrices

for accurate computation of the singular values.

AMS subject classification (2000): 65F15.

Key words: Hermitian matrix, eigenvalues, scaled diagonally dominant matrix, singu-

lar values, symmetric scaling, relative perturbations.

1 Introduction and notation.

Incentive for this research came with our intention to make a sound accuracyproof for the Kogbetliantz method. Since this is a two-sided Jacobi-like method,

we have tried to find a relative perturbation result for the singular values of a square matrix G, which expresses the bound by the condition cond(B), whereG = DBD and D is diagonal. The Kogbetliantz method typically starts witha triangular G, which is obtained after preprocessing by one or two QR factor-izations, as is advocated in [5] and [6]. Such G is more diagonal than beforepreprocessing. So, we looked for some perturbation result for the singular valuesof scaled diagonally dominant (s.d.d.) matrices.

In their pioneering paper on s.d.d. matrices, Barlow and Demmel [1] con-sidered symmetric scaling, but only for the eigenvalue problem of symmetric

matrices. From [3] and from the overview papers [13, 25], one finds out that for Received March 22, 2008. Accepted in revised form October 28, 2008. Communicated by

Axel Ruhe. This work was supported by the Croatian Ministry of Science, Education and Sports, grant

037-0372783-3042.



770 J. MATEJAS AND V. HARI

the relative perturbations of the singular values, the known results use eitherthe one-sided scaling or the general two-sided scaling G = D1BD2, combinedwith the assumptions on (all minors of) B which guarantee the existence of

a rank-revealing decomposition of G. The well behaved matrices for the singularvalue computation are also the matrices which satisfy some analytic conditionsor some sparsity and sign pattern, as well as some rationally structured matricesand some finite element matrices.

So, we tried to derive a new result for s.d.d. square matrices.We have started with the eigenvalue results from [1, Theorem 2, Proposition 4].

We have discovered that we can relax the assumptions and can get rid of thefactor n (as well as of the exponential function) in the bounds for the relative per-turbations of the eigenvalues of indefinite s.d.d. symmetric matrices. The proof uses a new technique which can be applied to similar perturbation problems.The result extends to some other classes of scaled diagonally dominant matri-ces, such as skew-Hermitian and hidden Hermitian matrices, the latter havingthe form D1HD2 with H Hermitian and D1, D2 diagonal, such that D1D2 ispositive definite.

With this simpler and somewhat sharper result for the indefinite s.d.d. Hermit-ian matrices, using the standard technique with the Wielandt matrix, we havederived a new, equally simple result for the singular values. This result enlargesthe class of well-behaved matrices to those square s.d.d. matrices G which havethe form DBD with D diagonal and B of small condition.

Although, the results presented here have their own significance, typical ap-plications lie in the accurate computation of the eigenvalues of s.d.d. Hermitianmatrices and of the singular values of s.d.d. square matrices. In particular, noneof the two approaches elaborated in [24, 27, 26] and in [4] for accurate compu-tation of the eigenvalues of an indefinite Hermitian matrix H is needed if H iss.d.d. The simple two-sided Jacobi method will accurately deliver the eigenvalues(see [17]). As our preliminary results imply, we are confident that for the accu-rate singular value computation of s.d.d. triangular matrices, the Kogbetliantzmethod will be excellent.

Throughout the paper, we use the following notation. By Cn×n is denotedthe set of n × n complex matrices and by Cn the set of complex column-vectorswith n components. For any square matrix X , diag(X ) stands for the diagonalpart of X , and Ω(X ) = X − diag(X ) for the off-diagonal part of X . By X and X F we denote the spectral and the Frobenius (Euclidean) matrix normof X , respectively. The Euclidean vector norm is also denoted by · . If notspecified otherwise, in this paper, the default choice of norm is the spectralnorm. For a Hermitian matrix H , λi(H ) denotes the i-th largest eigenvalueof H . The largest and the smallest eigenvalue of H are also denoted by λmax(H )and λmin(H ). Similarly, σi(G) denotes the i-th largest singular value of G, while

σmax(G) and σmin(G) denote its largest and smallest singular value. The absolutevalue of X = (xij) is the matrix |X | = (|xij |).

The paper is organized as follows. In Section 2 we briefly recall the definition of scaled diagonally dominant matrices. In Section 3, we prove the perturbation the-orem for indefinite s.d.d. Hermitian matrices, make comparison with the existing



RELATIVE PERTURBATIONS OF SDD MATRICES 771

result and present its immediate corollaries. In Section 4, we extend the resultto the relative perturbations of the singular values of s.d.d. square matrices.

2 Scaled diagonally dominant matrices.

Here, we recall the notion of scaled diagonally dominant matrices (see [1]).Let G ∈ Cn×n, G = (gij). Then G is α-diagonally dominant with respect to

a norm · if Ω(G) ≤ α min1≤i≤n |gii|, with 0 ≤ α < 1. Now, let B ∈ Cn×n

with |bii| = 1, 1 ≤ i ≤ n and let DL, DR be arbitrary nonsingular diagonalmatrices. Then G = DLBDR is α-scaled diagonally dominant (α-s.d.d.) withrespect to a given norm, if B is α-diagonally dominant with respect to thatnorm. If G is Hermitian (i.e. G = G∗), it is presumed that DL = D∗

R. Note that

an α-s.d.d. matrix has nonzero diagonal elements.In this paper, we consider the two-sided scaling under the constraint D∗L =

DR = D, where D is chosen to make the absolute values of the diagonal elementsof B = D−∗GD−1 one. For such scaling the notion of the scaled diagonallydominant matrix is defined in the same way as for the Hermitian matrix.

Let H ∈ Cn×n be a Hermitian matrix with the non-zero diagonal elements.Then the eigenvalues and the diagonal elements of H are real and

A = |diag(H )|−1/2H |diag(H )|−1/2

is the scaled matrix H . The diagonal elements of A are ones or minus ones. If Ω(A) ≤ α < 1, then H is α-s.d.d. Since α < 1, A and consequently H mustbe nonsingular.

Each α-s.d.d. matrix has a special structure (see [9, 18, 8]). This structure hasimpact on the rate of convergence of the appropriate diagonalization methods(see [15, 16, 19, 10, 20, 23]). Here we consider only the perturbation propertiesof the eigenvalues and singular values of such matrices.

3 Perturbation results for s.d.d. Hermitian matrices.

Let H ∈Cn×n be a Hermitian nonsingular matrix and δH a Hermitian perturb-ation of H . Let λ1 ≥ λ2 ≥ · · · ≥ λn and λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvaluesof H and H + δH , respectively. Then the standard perturbation theory (see [22])yields the following result:

If δH ≤ ηH , η < 1, then

λi − λiλi

≤ ηH |λi| ≤ ηκ(H ), 1 ≤ i ≤ n,

where κ(H ) = H ·H −1 is the condition number of H . The following strongerresult has been proven for the positive definite matrices in [2]:

If δA ≤ η < σmin(A), then

λi − λiλi

≤ η

σmin(A)≤ ηκ(A), 1 ≤ i ≤ n,(3.1)

where A = D−1HD−1, δA = D−1δH D−1 and D = [diag(H )]1/2.




Since almost diagonal Hermitian matrices have many properties common topositive definite ones, the question arises whether a similar result to (3.1) holdsfor α-s.d.d. indefinite Hermitian matrices. The answer is given in [1, Prop-

osition 4, Theorem 4] and here is our improvement of these results.Theorem 3.1. Let H and δH be Hermitian matrices of order n and let λ1 ≥

λ2 ≥ · · · ≥ λn and λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of H and H + δH ,respectively. Let D = |diag(H )|1/2 be nonsingular and let A = D−1HD−1, δA =D−1δH D−1. Let α, η be real numbers such that Ω(A) ≤ α and δA ≤ η.

If η + 2α < 1 then

λiλi

− 1

≤ η

1 −

1 + η1−2α

α≤ η

1 − 2α, 1 ≤ i ≤ n.(3.2)

If δH = 0, then the first inequality is strict provided that η > δA or α > 0. If η > 0, the second inequality is strict if and only if α > 0.

Proof. Let

µ =η

1 − 2α, ρ =

η

1 − (1 + µ)α.(3.3)

The assumption η +2α < 1 implies 2α < 1, so µ is well defined. It further implies

µ < 1 and ρ

≤µ,

which proves the second inequality in the assertion of the theorem. Note thatρ = µ if and only if α = 0 or η = 0, so ρ < µ if and only if H is not diagonaland δH = 0.

Since generally, for ε > 0,a

b− 1

≤ ε if and only if [1 − sgn(b)ε]b ≤ a ≤ [1 + sgn(b)ε]b, b = 0,

we have to prove the inequalities

[1 − sgn(λi)ρ] λi ≤ λi ≤ [1 + sgn(λi)ρ]λi, 1 ≤ i ≤ n.(3.4)

To avoid confusion about using ρ and µ, we note that µ serves only to simplifythe expression of the bound in the relation (3.2). In fact, we could have used µinstead of ρ in (3.4). But using it throughout in the proof, would result in thesecond (weaker) estimate in (3.2), |λi/λi − 1| ≤ µ. Since ρ ≤ µ, we proceedwith (3.4) to obtain the first (sharper) estimate |λi/λi − 1| ≤ ρ.

The assumption δA ≤ η is equivalent to the condition

|x∗δH x

| ≤ηx∗D2x, x

∈Cn.

Indeed, this claim follows from the following consideration. Note that

|y∗δAy|y∗y

=|x∗δH x|x∗D2x

, y = Dx, x = 0.




Since D2 is positive definite, y runs through the whole Cn \ 0 if and onlyif x does the same. Hence the both ratios assume the same values, and for somevalues of x and y they assume the maximum value which is δA.

Hencex∗(H − ηD2)x ≤ x∗(H + δH )x ≤ x∗(H + ηD2)x, x ∈ Cn.

Recalling the monotonicity property of the eigenvalues of Hermitian matrices,one easily obtains

λi(H − ηD2) ≤ λi ≤ λi(H + ηD2), 1 ≤ i ≤ n.(3.5)

The inequalities are attainable for δH = ±ηD2. Here we have assumed the non-increasing ordering of the eigenvalues, as it has been noted in the introduction.

The rest of the proof is based on the following idea.If H is positive definite, then in the transition from H to H ±ηD2, the diagonal

elements are shifted by the factor 1 ± η. We shall construct the positive semidef-inite matrix M such that in the transition from H to H ±ηD2+M the eigenvaluesare shifted by the factor 1±ρ. This is achieved by setting M = (ρ−η)D2 +ρΩ(H )(see Corollary 3.3). Then we have H ± ηD2 + M = (1 ± ρ)H .

If H is Hermitian indefinite, then in the transition from H to H + ηD2, thepositive (negative) diagonal elements are shifted by the factor 1 + η (1 − η).Similarly, in the transition from H to H − ηD2, the negative (positive) diagonalelements are shifted by 1 + η (1 − η). In these cases the matrix M is constructed

according to the block partition of H as follows.We can assume that the diagonal elements of H satisfy

h11 ≥ h22 ≥ · · · ≥ hmm > 0 > hm+1,m+1 ≥ · · · ≥ hnn.

Otherwise, we can consider the matrix P T HP , where P is an appropriate per-mutation matrix. So, m > 0 and n−m > 0 are numbers of positive and negativediagonal elements, respectively. According to the partition of n, n = m+(n−m),we define the block-matrix partition

H = H 11 H 12

H ∗12 H 22 , A = A11 A12

A∗12 A22 , D = D1 O

O D2 ,

where H 11, A11, D1 ∈ Cm×m and H 22, A22, D2 ∈ C (n−m)×(n−m). Note thatdiag(H ) = diag(D2

1 , −D22). The Hermitian matrix M is constructed in the fol-

lowing way

M = (ρ − η)D2 +

ρΩ(H 11) (

1 − ρ2 − 1)H 12

(

1 − ρ2 − 1)H ∗12 −ρΩ(H 22)

.(3.6)

Let us inspect whether M is positive semidefinite. We have

Ω(M ) = ρI m OO −ρI n−m

· Ω(H 11) OO Ω(H 22)

− ρ2

1 +

1 − ρ2

O H 12

H ∗12 O

,




and using the relation (3.3), we obtain

Ω(D−1M D−1)

≤ρ

· Ω(A11) O

O Ω(A22) +

ρ2

1 +

1 − ρ2· O A12

A∗12 O

≤ ρα + ρ2α = ρα(1 + ρ) ≤ ρα(1 + µ)

= ρ[1 − 1 + (1 + µ)α] = ρ − η.

Since diag(D−1M D−1) = (ρ − η)I , we conclude that D−1M D−1 is positivesemidefinite. The law of inertia implies that M is also positive semidefinite. We

also see that M is positive definite as soon as α > 0 and η > 0, that is, as soonas δH = 0 and H is not diagonal.

Next, we consider the matrix H + ηD2 + M . Since M is positive semidefinite,we have

λi(H + ηD2) ≤ λi(H + ηD2 + M ), 1 ≤ i ≤ n.(3.7)

Here, we have strict inequalities provided that δH = 0 and H is not diagonal.It is easy to see that

H + ηD2 + M = (1 + ρ)H 11 1 − ρ2H 121 − ρ2H ∗12 (1 − ρ)H 22

= ∆H ∆,(3.8)

where

∆ =

√1 + ρI m O

O√

1 − ρI n−m

is positive definite. Note that λ1(∆2) = 1 + ρ, λn(∆2) = 1 − ρ. Recalling theperturbation theorem of Ostrowski (see [12, Theorem 4.5.9]), we obtain

λi(∆H ∆) = θiλi(H ) ≤

(1 + ρ)λi if λi > 0

(1 − ρ)λi if λi < 0, 1 ≤ i ≤ n.(3.9)

Combining the relations (3.5), (3.7), (3.8) and (3.9), we obtain

λi ≤ λi(∆H ∆) ≤ [1 + sgn(λi)ρ]λi, 1 ≤ i ≤ n,

which is the second inequality in the assertion (3.4).The proof of the first inequality in (3.4) uses a similar series of conclusions,

based on the relations

M = (ρ − η)D2 +

ρΩ(H 11) (1 −

1 − ρ2)H 12

(1 −

1 − ρ2)H ∗12 −ρΩ(H 22)

(3.10)




and

H

−ηD2

− M = (1 − ρ)H 11

1 − ρ2H 12

1 − ρ2

H ∗

12 (1 + ρ)H 22 = ∆H ∆,

where

∆ =

√1 − ρI m O

O√

1 + ρI n−m

.

3.1 Comparison with bound of Barlow and Demmel.

Our bound in Theorem 3.1 is sharper than the bound [1, Proposition 4] by

a factor n.In our notation [1, Theorem 4] reads:

Let H be symmetric γ -s.d.d. matrix with respect to the 2-norm. Assume that

K ξ = H + ξδH is γ -s.d.d. for all 0 ≤ ξ ≤ 1. Then

e

−η

1−γ ≤ λiλi

≤ e

η

1−γ implying

λiλi

− 1

≤ e

η

1−γ − 1 , 1 ≤ i ≤ n.(3.11)

We have the following observations.

1. Let Aξ

=|diag(K

ξ)|−1/2K

ξ|diag(K

ξ)|−1/2. Then A

0= A, where A is from

Theorem 3.1, and let α = Ω(A). We always have

α = Ω(A0) ≤ γ.

If we assume that γ = max0≤ξ≤1 Ω(Aξ), then γ is not simple to compute.Hence for given H and δH , one can use a value for γ which is an overestimateof max0≤ξ≤1 Ω(Aξ). This further increases the distance between α and γ .

2. Let µ = η/(1 − 2α) < 1 and γ < 1. Then

φ =e

η

1−γ

− 1 =

η

1 − γ +

η2

2(1 − γ )2 +

η3

6(1 − γ )3 + · · ·ρ =

η

1 − (1 + µ)α=

η

1 − α+ r, r =

η2α

(1 − α)(1 − 2α)[1 − (1 + µ)α].

If η = 0, then ρ = φ = 0. If η > 0 and α = 0, then ρ < φ.Let η > 0, α > 0 and let us estimate for what α, ρ < φ holds. Since α ≤ γ ,we have η/(1 − α) ≤ η/(1 − γ ). Now, if r ≤ η2/[2(1 − γ )2] then ρ < φ willhold. Therefore, we estimate the ratio

r :

η2

2(1 − γ )2 ≤ r :

η2

2(1 − α)2 =

2α(1

−α)

(1 − 2α)[1 − (1 + µ)α] <

2α(1

−α)

(1 − 2α)2 .

If the value of this ratio is not greater than 1 then we have ρ < φ. Oneeasily finds out that 2α(1 − α) ≤ (1 − 2α)2 holds whenever α ≤ α0, α0 =(3 − √

3)/6 ≈ 0.2113.




Therefore, for any α-s.d.d. Hermitian matrix H with 0 < α ≤ α0, and forany Hermitian δH satisfying 0 < δA ≤ η < 1 − 2α, we have ρ < φ, whichmeans that the estimate of Theorem 3.1 is sharper than the corresponding

estimate of [1, Theorem 4].But if α is not close to γ , the same conclusion can hold for α > α0. Generally,if α > α0, then the relation between ρ and φ may depend on the structureof the perturbation δH , i.e. sometimes one is larger than the other or viceversa, i.e. the comparison between the two bounds is not clear.

3.2 Consequences.

Let us note that D in Theorem 3.1 can be taken complex. Indeed, let D =DH Φ, where DH = |diag(H )|1/2 and Φ is diagonal and unitary. It then suffices

to apply Theorem 3.1 to Φ∗

H Φ and Φ∗

δH Φ and use the relations

Ω(Φ∗H Φ) = DH Ω(Φ∗AΦ)DH , Φ∗AΦ = A, λi(H ) = λi(Φ∗H Φ),

Φ∗δH Φ = DH (Φ∗δAΦ)DH , Φ∗δAΦ = δA, λi(H ) = λi(Φ∗H Φ),

where 1 ≤ i ≤ n. Here, H = H + δH . The same redefinition of D can be madein the ensuing corollaries.

If H and δH are skew-Hermitian (H ∗ = −H , δH ∗ = −δH ), then Theorem 3.1can be applied to −ıH and −ıδH , which are Hermitian. Here ı =

√−1. Since themultiplication by

−ı does not affect the norms, the statement of Theorem 3.1

trivially holds for skew-Hermitian matrices. If A is a real skew-symmetric matrixof even order, which is almost in Murnagham form (see [21]), then −ıA has tobe transformed by n/2 complex Jacobi rotations in the planes (2 j − 1, 2 j). Then±a2j−1,2j become diagonal elements and the results are obtained from the soobtained Hermitian matrix.

Our first corollary has an important application in connection with the accur-acy of the Jacoobi method for indefinite symmetric matrices (see [17]). In thisapplication η is bounded by a small multiple of Ω(A)F , where A is the scalediteration at the beginning of the sweep (see [17, Remark 11]). As the process

advances, α = Ω(A)F tends to zero, so in the later stage of the process α 1and η < α. The estimate (3.11) can be applied too, but as explained above, inthis application Theorem 3.1 will be sharper. However, Corollary 3.2 below, ismost appropriate to apply, since Ω(A)F is easier to compute than Ω(A) andin addition, η is bounded by a multiple of Ω(A)F , and finally Corollary 3.2yields sharper estimate then Theorem 3.1.

Corollary 3.2. If, in Theorem 3.1, the condition Ω(A) ≤ α is replaced by Ω(A)F ≤ α, then η + α < 1 implies |λi/λi − 1| ≤ η/(1 − α), 1 ≤ i ≤ n.

Proof. In the proof of Theorem 3.1, we have ρ = η/(1−

α) < 1 and we donot use µ. Using the relation (3.6), we easily obtain

Ω(D−1M D−1)F ≤ max

ρ,

ρ2

1 +

1 − ρ2

· Ω(A)F ≤ ρα = ρ − η.




By help of (3.10), the same estimate is obtained for M . Hence M and M arepositive semidefinite. The rest of the proof remains the same.

In the case of definite Hermitian matrices, the assertion of the theorem canbe further improved to yield the bound which is similar, although somewhatweaker than the bound in [2]. Namely, in (3.1), σmin(A) is replaced by 1 −σmax(Ω(A)).

Corollary 3.3. If, in Theorem 3.1, H is (positive or negative) definite, then η + α < 1 implies

λiλi

− 1

≤ η

1 − α, 1 ≤ i ≤ n.(3.12)

Proof. We can assume that H is positive definite, otherwise we con-sider −H . In the proof of Theorem 3.1 we set µ = ρ = η/(1 − α) < 1. Since nowdiag(H ) = D2, instead of using the relations (3.6) and (3.10), we define

M = M = (ρ − η)D2 + ρΩ(H ).

We have

Ω(D−1M D−1) = ρΩ(A) ≤ ρα = ρ − η,

which implies that M and M are positive semidefinite. Next, we have

H + ηD2 + M = (1 + ρ)H = ∆H ∆, ∆ =

1 + ρI,

H − ηD2 − M = (1 − ρ)H = ∆H ∆, ∆ =

1 − ρI.

The rest of the proof follows the lines of the proof of Theorem 3.1.

The following corollary gives the relative perturbation result for the “hiddenHermitian” matrices, i.e. those of the form D1HD2 where D1 and D2 are diag-

onal and nonsingular.

Corollary 3.4. Let K = D1HD2 ∈ Cn×n, δK = D1δH D2 ∈ Cn×n, whereH and δH are Hermitian and D1, D2 diagonal such that D1D2 is positive def-inite. Then K and K + δK have real eigenvalues. Let λ1 ≥ λ2 ≥ · · · ≥ λnand λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of K and K + δK , respectively. Let D = |diag(H )|1/2 be nonsingular and let A = D−1HD−1, δA = D−1δH D−1. Let α, η be real numbers such that Ω(A) ≤ α and δA ≤ η. If η +2α < 1, then therelation (3.2) holds. If the condition Ω(A) ≤ α is replaced with Ω(A)F ≤ α,then η + α < 1 implies the relation (3.12). The same conclusion holds provided

that H is definite and η + α < 1, where again Ω(A) ≤ α.

Proof. Let D1 = |D1|Φ, D2 = |D2|Φ∗, where Φ is diagonal and unitary. Let

∆ = |D1|− 1

2 |D2|1/2, D3 = [D1D2]1/2 = |D1|1/2|D2|1/2 = |D1|∆,




and let N = ∆K ∆−1 and δN = ∆δK ∆−1. Then the eigenvalues of N andN + δN are λi and λi, 1 ≤ i ≤ n. Since

N = ∆K ∆−1

= |D1|−1/2

|D2|1/2

(D1HD2)|D1|1/2

|D2|−1/2

= D3ΦH Φ∗

D3

= D3D(ΦAΦ∗)DD3 = D4AΦD4,

δN = ∆δK ∆−1 = |D1|−1/2|D2|1/2(D1δH D2)|D1|1/2|D2|−1/2 = D3ΦH Φ∗D3

= D3D(ΦδAΦ∗)DD3 = D4δAΦD4,

where AΦ = ΦAΦ∗, δAΦ = ΦδAΦ∗ and D4 = DD3. Since for any unitary invari-ant norm AΦ = A and δAΦ = δA, the assertions of the corollary followby applying Theorem 3.1, Corollary 3.2 and Corollary 3.3 to N and N + δN .

4 Relative perturbations of the singular values.

Here, we derive new relative perturbation estimates for the singular values of an α-s.d.d. square matrix G.

Theorem 4.1. Let G and δG be square matrices of order n and let σ1 ≥ σ2 ≥· · · ≥ σn and σ1 ≥ σ2 ≥ · · · ≥ σn be the singular values of G and G = G + δG,

respectively. Let D = |diag(G)|1/2

be nonsingular and let B = D−1

GD−1

, δB =D−1δGD−1. Let α, η be real numbers such that Ω(B) ≤ α and δB ≤ η.If η + 2α < 1, then G and G are nonsingular and σi

σi− 1

≤ η

1 − 1 + η

1−2α

α

≤ η

1 − 2α, 1 ≤ i ≤ n.

If δG = 0, then the first inequality is strict provided that η > δB or α > 0. If η > 0, the second inequality is strict if and only if α > 0.

Proof.First, let us show that G and G

are nonsingular. Indeed, since

σmin(B) ≥ 1 − α > 0 and σmin(B + δB) > 1 − α − η > α ≥ 0,

B and B = B + δB are nonsingular. Hence G = DBD and G = DBD asproducts of three nonsingular matrices are nonsingular. The rest of the proof uses the technique from the proof of [18, Lemma 2]. However, instead of frequentreferring the lines of that proof, we shall provide here a complete proof of thetheorem.

Note that Φ∗G, where Φ = diag(ei arg(g11), . . . , ei arg(gnn)), has nonnegative

diagonal elements. Let G = Φ∗

G and G

= Φ∗

G

. Now, consider the unitarymatrix

U =

Φ 00 I n

1√

2

I n −I nI n I n

=

1√2

Φ −ΦI n I n




and the Hermitian matrices

H = U ∗ 0 G

G∗

0 U =1

2 G +

G∗

G −

G∗

−( G − G∗

) −( G + G∗

) ,(4.1)

H = U ∗

0 G

[G]∗ 0

U =

1

2

G + [ G]∗ G − [ G]∗

−( G − [ G]∗) −( G + [ G]∗)

.(4.2)

The eigenvalues of H are (cf. [7, Section 8.6]) σ1 ≥ · · · ≥ σn > −σn ≥ · · · ≥ −σ1

and the diagonal elements are |gii|, 1 ≤ i ≤ n in the first n positions and −|gii|,1 ≤ i ≤ n in the last n positions on the diagonal. The eigenvalues of H areσ1 ≥ · · · ≥ σn > −σn ≥ · · · ≥ −σ1. Let us now show that H is α-s.d.d. if andonly if G is α-s.d.d. To this end it suffices to prove

Ω(A) = Ω(B), H = ∆A∆, ∆ = |diag(H)|1/2.(4.3)

Obviously,

∆ =

D 00 D

.

A straightforward calculation yields ∆U = U ∆. Hence

A = ∆−1H ∆−1 = U ∗∆−1U U ∗ O GG∗ O U U ∗∆−1U = U ∗ O BB∗ O U.

This agrees with the fact that diag(Φ∗B) = I , which holds because Φ∗ and Dcommute. An inspection of the relation (4.1) reveals that

Ω (A) = U ∗

O Ω (B)Ω (B)∗ O

U

which implies (4.3).

It remains to prove that δA ≤ η, where δA = ∆−1

(H

−H )∆−1

, and to applyTheorem 3.1 to H and H . But this follows from the relation (4.1) and (4.2),since

U ∆−1δH ∆−1U ∗ =

0 δB

δB∗ 0

and

0 δBδB∗ 0

= δB ≤ η.

Note that using Corollary 3.2 instead of Theorem 3.1 yields the implication

√2α + η < 1 ⇒ σi

σi − 1 ≤η

1 − √2α , 1 ≤ i ≤ n, where Ω(B)F ≤ α.

Finally, using the argument from the first paragraph of Section 3.2, in The-orem 4.1 one can use any complex diagonal D such that G = D∗BD andδG = D∗δBD and |bii| = 1 for all 1 ≤ i ≤ n.




A typical application of Theorem 4.1 lies in obtaining sharp accuracy estimatesof the Kogbetliantz method for computing the SVD of triangular matrices. Asmentioned in the introduction, after one or two initial QR factorizations, the

obtained matrix will be more diagonal than the starting one. Applying after-wards the Kogbetliantz method, the iterated matrix will soon become scaleddiagonally dominant. In [11] sharp accuracy estimates have been derived for therotational parameters and for the updated diagonal elements corresponding toone Kogbetliantz step. With these results, which include the case of a generaland of an α-s.d.d. triangular matrix, one can obtain sharp accuracy estimatesfor one step and for one sweep of the method. In the early stage of the processthe proof uses the technique from [14]. But in the later stage, when the iteratedmatrix becomes α-s.d.d., it is appropriate to make the estimates dependent on αwhich tends to zero. At this stage, the analysis resembles to the one in [17] andTheorem 4.1 is used.

We have used MATLAB to see how good are the bounds in Theorems 3.1and 4.1. The tests confirmed that they are good upper estimates of the realchange of the eigenvalues and singular values.

Acknowledgement.

The authors are thankful to the anonymous referees for their helpful sugges-tions how to make the paper better.

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