The palindromic generalized eigenvalue problem A ∗ x = λ Ax: Numerical solution and applications

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Linear Algebra and its Applications

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / l a a

The palindromic generalized eigenvalue problem A ^∗ x = λ Ax: Numerical solution and applications

Tiexiang Li

^a

, Chun-Yueh Chiang

^b

, Eric King-wah Chu

^c,∗

, Wen-Wei Lin

^d

aDepartment of Mathematics, Southeast University, Nanjing 211189, People’s Republic of China bCenter for General Education, National Formosa University, Huwei 632, Taiwan

cSchool of Mathematical Sciences, Building 28, Monash University, VIC 3800, Australia

dCMMSC and NCTS, Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan

A R T I C L E I N F O A B S T R A C T

Article history:

Received 29 June 2009 Accepted 19 November 2009 Available online 12 January 2010 Submitted by V. Mehrmann

AMS classification:

15A18 15A22 65F15

Keywords:

Palindromic generalized eigenvalue problem

Doubling algorithm Singular descriptor system

In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A^∗x=λAx. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal control for singular descriptor linear systems will be presented to illustrate the feasibility and efficiency of the PDA.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

In this paper, we develop the palindromic doubling algorithm (PDA) for the numerical solution of the palindromic generalized eigenvalue problem (PGEP)

∗Corresponding author. Tel.: +61 3 99054480; fax: +61 3 99054403.

E-mail addresses: [email protected] (T. Li), [email protected] (C.-Y. Chiang), [email protected] (E.K.-w. Chu), [email protected] (W.-W. Lin).

0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved.

doi:10.1016/j.laa.2009.12.020

A^∗x

= λ

^{Ax, (1.1)} whereAis a real or complexN

×

^Nmatrix,

λ ∈

C^andx

∈

C^N

\{

⁰

}

are eigenvalue and the correspond- ing eigenvector of (1.1), respectively. Here, the symbol “

∗

^”

=

(Transpose) or H (Hermitian). The pencil A^∗

− λ

^Aand the pair

(

^{A∗, A}

)

are usually called a palindromic linear pencil and a palindromic matrix pair, respectively. It is easily seen that the eigenvalues of (1.1) satisfy the reciprocal property, i.e., they appear in pairs as in

{λ

^{, 1}

/λ

^∗

}

^.

The PGEPs with complex coefficient matrices were firstly suggested as “good” linearizations [5,6]

of palindromic polynomial/quadratic matrix pencils, arising from the study of vibration analysis [2, 4,12]. A PGEP with real coefficient matrices can also be shown to be equivalent to the generalized continuous/discrete-time algebraic Riccati equations, associated with the continuous/discrete-time, linear-quadratic optimal control problems (see [11] for details).

The standard approach for solving the PGEP is to compute its generalized Schur form (e.g., byqzin MATLAB), ignoring its symmetric or palindromic structure in

(

^{A∗, A}

)

. However, the reciprocal property of eigenvalues of (1.1) is not preserved by computation generally, producing large numerical errors [7].

Recently, a QR-like algorithm [8] and a hybrid method [7] (which combines Jacobi-type method with the Laub’s trick) were proposed for the PGEP. The QR-like algorithm generally requiresO

(

^N4

)

^flops and the hybrid method requiresO

(

^N3log

(

^N

))

flops. Alternatively, for methods of cubic complexity, a URV-decomposition based structured method [9] and a structure-preserving algorithm [3] for PGEPs were proposed, producing eigenvalues which are paired to working precision. Unfortunately for PGEPs, these more efficient (and equivalent) methods require the transformation of the PGEP to the quadratic form

(μ

^2A∗

+ μ ·

⁰

+

^A

)

^x

=

0, leading to operations in larger 2N

×

^2Nmatrices. The PDA is a unique and more direct, thus more efficient, algorithm for the PGEP.

The purpose of this paper is to develop the PDA for solving the PGEP structurally. We establish quadratic convergence and linear convergence with rate 1

/

2 of the PDA, respectively, when

(

^{A∗, A}

)

^has no unimodular eigenvalues and has unimodular eigenvalues with partial multiplicities two. In appli- cation to discrete-time optimal control problems, we especially develop a new algorithm combined with the PDA (as in Algorithm 4.1) for solving the optimal control ofsingulardescriptor linear systems.

To our knowledge, the associated generalized discrete-time algebraic Riccati equation (GDARE) has not been solved successfully in a structure-preserving manner.

This paper is organized as follows. In Section 2 we develop the palindromic doubling algorithm (PDA) for solving PGEPs. In Section 3 we establish the convergence theory for the PDA. In Section 4 we use the PDA to compute numerical solutions structurally in different applications in PGEPs, GCAREs and GDAREs. Concluding remarks are given in Section 5.

Throughout this paper,C^{m×n and} R^m×n denote the sets ofm

×

ⁿ complex and real matrices, respectively. For convenience, we denoteCⁿ

=

C^n×1,C

=

C^1,Rⁿ

=

R^n×1andR

=

R^{1. The open} left-half plane and the open unit disk are denoted byC₋^andD1, respectively; 0m×ⁿ

(

⁰m

)

^andImare them

×

ⁿ

(

^m

×

^m

)

zero matrix and them

×

^midentity matrix, respectively. We use

σ(

^{A, B}

)

^{to denote} the spectrum of the matrix pair

(

^{A, B}

)

^{, and}

·

denotes the 2-norm of a matrix.

2. Palindromic doubling algorithm

For a given palindromic matrix pair

(

^{A∗, A}

)

, we shall develop a doubling algorithm for solving the associated PGEP which preserves the palindromic structure at each iterative step.

Suppose

−

¹

∈ / σ(

^{A∗, A}

)

(the assumption can be removed later in Remark 3.1). We then have A^∗

(

^A∗

+

^A

)

^−1A

= ((

^A∗

+

^A

) −

^A

)(

^A∗

+

^A

)

⁻¹

(

^A

+

^A∗

−

^A∗

)

= (

^I

−

^A

(

^A∗

+

^A

)

⁻¹

)((

^A∗

+

^A

) −

^A∗

)

=

^A

(

^A∗

+

^A

)

^−1A∗

.

^(2.1)

From (2.1), it is easily seen that

A

(

^A∗

+

^A

)

^{−1, A∗}

(

^A∗

+

^A

)

⁻¹

−

^A∗A

=

⁰

.

^(2.2)

We now define the doubling transformA

→

^Aby

A

=

^A

(

^A∗

+

^A

)

^−1A

.

^(2.3)

Theorem 2.1.The matrix pair

(

^A∗,A

)

has the doubling property

;

ⁱ

.

^e

.

^{, if}

A^∗U

=

^{AUS, (2.4)}

where U

∈

C^{N×and S}

∈

C^{×, then}

A^∗U

=

^AUS2

.

^(2.5)

Proof.Multiplying the both sides of (2.4) byA^∗

(

^A∗

+

^A

)

⁻¹, and (2.1) and (2.4) imply (2.5).

From Theorem2.1, we see that the doubling transform (2.3) preserves the palindromic structure.

So, for a palindromic matrix pair A^∗₀, A0

withA0

∈

C^N×N, we can develop the PDA to generate the sequence A^∗_k, Ak

if no breakdown occurs in the iterative process.

PDA AlgorithmGivenA₀

∈

C^N×N,

τ

(a small tolerance), fork

=

^0,1,2,

. . .

^,compute

A_k+1

=

^Ak

A^∗_k

+

^Ak

₋1

A_k, (2.6)

ifdist

(

^Null

(

^Ak+¹

)

^,Null

(

^Ak

)) < τ

, then stop, end for

Here, “Null

(·)

” denotes the null space of the given matrix and “dist

(·

^,

·)

” denotes the distance between two subspaces.

To develop the PDA further, denote

A_k

=

^Hk

+

^Kk, (2.7a)

where H_k

=

¹

2

A^∗_k

+

^Ak

=

^H∗k, K_k

=

¹ 2

A_k

−

^A∗k

= −

^Kk^∗ (2.7b)

are the

∗

-symmetric and

∗

-anti-symmetric parts ofAk, respectively. Then the iteration (2.6) can be rewritten as

A_k+1

=

^Hk+¹

+

^Kk+¹

=

¹

2

(

^Hk

+

^Kk

)

^Hk⁻¹

(

^Hk

+

^Kk

)

=

¹ 2

H_k

+

^KkH_k⁻¹K_k

+

^Kk

.

The iteration (2.6) in the PDA can be simplified to H_k+1

=

¹

2

H_k

+

^K0H_k⁻¹K₀ , K_k+1

=

^Kk

= · · · =

^K0

.

3. Convergence of PDA

LetA0

∈

C^N×N. Suppose the eigenvalue “1" of A^∗₀, A0

(if exists) has partial multiplicity one or two, and the other unimodular eigenvalues of

A^∗₀, A0

(if exist) have exactly partial multiplicities two. By the theorem of Kronecker canonical form there are nonsingular matricesQandZsuch that

QA^∗₀Z

=

J0

⊕

Ω0 0_,˜

⊕

^Ir

0_n,n˜ I_˜

⊕

Ω0

≡

^C0, (3.1a)

QA₀Z

=

₀^{In 0n,n˜}

˜

n,n J₀^∗

⊕

^Ir

≡

^D0, (3.1b)

whereΩ0

=

^diageiω1,

. . .

^{, eiωr,J}0

∈

C^×consists of stable Jordan blocks (i.e.,

ρ(

^J0

) <

^{1, where}

ρ(·)

is the radius of the spectrum) andJ₀^∗

=

^J0

⊕

^Imwithn

= +

^r,

˜ = +

^m,n

˜ =

ⁿ

+

^m

= +

r

+

^mandN

=

²ⁿ

+

^{m. Here “}

⊕

" denotes the direct sum of matrices.

SinceC₀D₀

=

^D0C₀, from (3.1b) we have thatA^∗₀ZD₀

=

^A0ZC₀. From Theorem2.1and steps in the PDA, it follows that

A^∗_kZD²₀^k

=

^AkZC₀^2k

.

^(3.2)

Substituting (3.1b) into (3.2), we get A^∗_kZ

I_n 0_n,n˜ 0_n,n˜ J₀^∗2^k

⊕

^Ir

=

^AkZ

⎡

⎣J₀^2k

⊕

Ω0^2k 0_,˜

⊕

k

0_˜n,n I

⊕

Ω0^2k

⎤

⎦, (3.3)

whereΓk

=

^2kΩ0^2k−1. On the other hand, we can interchange the role of A^∗₀, A0

by considering the pair

A₀, A^∗₀

which has the same Kronecker structure as A^∗₀, A₀

. Therefore, there are nonsingularP andYsuch that

PA₀Y

=

J^∗₀

⊕

Ω0^∗ 0_,˜

⊕

^Ir

0_n,n˜ I_˜

⊕

Ω0^∗

≡

^E0, (3.4a)

PA^∗₀Y

=

^I₀^{n 0n,˜n}

˜

n,n J0

⊕

^Ir

≡

^F0, (3.4b)

SinceE₀F₀

=

^F0E₀, we deduce thatA₀YF₀

=

^A∗0YE₀. Using the similar arguments as in (3.2) and (3.3), we obtain

A_kY

In 0n

0n J₀^2k

⊕

^Ir

=

^A∗kY

⎡

⎣

J₀^∗2^k

⊕

Ω0^∗

2^k

0_,˜

⊕

Γk^∗

0_n,n˜ I_˜

⊕

Ω0^∗

2^k

⎤

⎦

.

^(3.5)

We partitionA_k,H_kandK₀in (2.7a) into four sub-blocks as in Ak

=

^A_A^{k1 Ak3}

k2 Ak4

, Hk

=

^H_H^{k1 H∗k2}

k2 Hk4

, K0

=

^K01

−

^K02^∗

K02 K04

, (3.6a)

whereAk1,Hk1,K01

∈

C^n×n,A∗k2, Ak3, H_k2^∗, K₀₂^∗

∈

C^nטnandAk4, Hk4, K04

∈

C^˜nטn. From (2.7a) and (3.6a), we also have

Ak1

=

^Hk1

+

^K01, Ak2

=

^Hk2

+

^K02, (3.6b)

Ak3

=

^H∗k2

−

^K02^∗, Ak4

=

^Hk4

+

^K04

.

^(3.6c)

Furthermore, we partitionZin (3.3) andYin (3.5) as in Z

=

^Z_Z^{1 Z3}

2 Z₄

, Y

=

^Y_Y^{1 Y3}

2 Y₄

, (3.7)

whereZ1,Y1

∈

C^n×n;Z∗2, Z3, Y₂^∗, Y3

∈

C^nטnandZ4, Y4

∈

C^n˜×˜n. For convenience, we denote

Zi,a

≡

^Zi

(:

^,1

: )

^{, Y}i,a

≡

^Yi

(:

^,1

: )

^{, i}

=

^{3,4, (3.8)} Z_i,b

≡

^Zi

(:

^,

+

¹

:

ⁿ

)

^{, Y}i,b

≡

^Yi

(:

^,

+

¹

:

ⁿ

)

^{, i}

=

^1,2

.

^(3.9)

Theorem 3.1.Let A₀

∈

C^N×N

.

Suppose that the eigenvalue “1" of A^∗₀, A₀

(

^{if exists}

)

has partial multiplicity one or two, the other unimodular eigenvalues of

A^∗₀, A0

(

^{if exist}

)

have exactly partial multiplicities two, and

(

^3.1b

)

^and

(

^3.4b

)

^{hold withn}

˜

2

(

ⁱ

.

^e

.

^{, r}

+

^m

).

Suppose that Z₁and Y₁in

(

^3.7

)

are invertible, and W

≡ [

^Z3,a

−

^Z4,a

|

^Y3,a

−

^Y4,a

] ∈

C^n˜×2is of full row rank, whereΦ

≡

^Z2Z₁⁻¹,Ψ

≡

^Y2Y₁⁻¹

.

^If

−

¹

∈ /

^rj=¹

e^2kiωj, k0

, then the sequence A^∗_k, A_k

generated by the PDA is well defined and satisfies A^∗_k

Z₁ Z₂

→

^0, linearly ask

→ ∞

^{, (3.10a)}

A_k Y1

Y₂

→

^0,linearly ask

→ ∞

^{, (3.10b)}

with convergence rate1

/

^{2, wherespanZ}_Z¹₂^andspanY_Y³₄form the weakly stable and the unstable invariant subspaces of

A^∗₀, A0

corresponding to

(

^J0

⊕

Ω0, In

)

^andIn, J₀^∗

⊕

Ω0^∗

, respectively

.

Proof.Since

−

¹

∈ /

^rj=¹

e^2kiωj, k0

, from (2.6) we see that

−

¹

∈ / σ

^A∗k, Ak

, thus,A^∗_k

+

^Akis in- vertible for allk.

From (3.6a), (3.3) and (3.7), we have A^∗_k1Z1

+

^A∗k2Z2

=

^Ak1Z1

J₀^2k

⊕

Ω0^2k

+

^Ak3Z2

J₀^2k

⊕

Ω0^2k

, (3.11)

A^∗_k3Z₁

+

^A∗k4Z₂

=

^Ak2Z₁

J₀^2k

⊕

Ω0^2k

+

^Ak4Z₂

J₀^2k

⊕

Ω0^2k

, (3.12)

A^∗_k1Z₃J₀^∗2^k

⊕

^Ir

+

^A∗k2Z₄J^∗₀2^k

⊕

^Ir

=

^Ak1

Z₁

(

⁰_,˜

⊕

Γk

) +

^Z3

I_˜

⊕

Ω0^2k

+

^Ak3

Z₂

(

⁰_,˜

⊕

Γk

) +

^Z4

I_˜

⊕

Ω0^2k

,(3.13) A^∗_k3Z3J₀^∗2^k

⊕

^Ir

+

^A∗k4Z4J₀^∗2^k

⊕

^Ir

=

^Ak2

Z1

(

⁰_,˜

⊕

Γk

) +

^Z3

I_˜

⊕

Ω0^2k

+

^Ak4

Z2

(

⁰_,˜

⊕

Γk

) +

^Z4

I_˜

⊕

Ω0^2k

.

^(3.14)

Post-multiplying (3.13) by 0_˜,

⊕

Γk⁻¹Ω0^2k, we get A^∗_k1Z3

0_˜,

⊕

Γk⁻¹Ω0^2k

+

^A∗k2Z4

0_˜,

⊕

Γk⁻¹Ω0^2k

= (

^Ak1Z1

+

^Ak3Z2

)

⁰

⊕

Ω0^2k

+ (

^Ak1Z3

+

^Ak3Z4

)

⁰_˜,

⊕

Ω0^2kΓk⁻¹Ω0^2k

.

^(3.15)

Substituting (3.15) into (3.11) and usingΩ0^2kΓk⁻¹Ω0^2k

=

^2−kΩ0^2k+1, we have A^∗_k1Z1

+

^A∗k2Z2

= (

^Ak1Z1

+

^Ak3Z2

)

^J0^2k

⊕

⁰r

+ (

^Ak1Z1

+

^Ak3Z2

)

⁰

⊕

Ω0^2k

= (

^Ak1Z1

+

^Ak3Z2

)

^J0^2k

⊕

⁰r

+

^A∗k1Z3

+

^A∗k2Z4 0_˜,

⊕

Γk⁻¹Ω0^2k

− (

^Ak1Z₃

+

^Ak3Z₄

)

⁰_˜,

⊕

^2−kΩ0^2k+1

.

^(3.16)

Using (3.6a) and re-arranging (3.16), we get H_k1Z

Z₁

I_n

−

^J0^2k

⊕

⁰r

−

^Z3

0_˜,

⊕

^2−kΩ0

I_r

−

Ω0^2k

+

^Hk2^∗

Z2

In

−

^J0^2k

⊕

⁰r

−

^Z4

0_˜,

⊕

^2−kΩ0

Ir

−

Ω0^2k

=

^K01

Z₁

I_n

+

^J0^2k

⊕

⁰r

−

^Z3

0_˜,

⊕

^2−kΩ0

I_r

+

Ω0^2k

−

^K02^∗

Z₂

I_n

+

^J0^2k

⊕

⁰r

−

^Z4

0_˜,

⊕

^2−kΩ0

I_r

+

Ω0^2k

.

^(3.17)

Denote

k

≡

^max

ρ(

^J0

)

^2k,2−k

→

^{0, ask}

→ ∞.

^(3.18)

SinceΩ0^2kis bounded andZ1is invertible, by lettingΦ

≡

^Z2Z₁⁻¹, (3.17) can be simplified to H_k1

= −

^H∗k2

(

Φ

+

^O

(

k

)) +

^K01

−

^K02^∗Φ

+

^O

(

k

).

^(3.19) Post-multiplying (3.13) byI_˜

⊕

Γk⁻¹, we have

A^∗_k1Z3J₀^∗2^k

⊕

Γk⁻¹

+

^A∗k2Z4J₀^∗2^k

⊕

Γk⁻¹

=

^Ak1

Z₁

(

⁰_,˜

⊕

^Ir

) +

^Z3

I_˜

⊕

Ω0^2kΓk⁻¹

+

^Ak3

Z₂

(

⁰_,˜

⊕

^Ir

) +

^Z4

I_˜

⊕

Ω0^2kΓk⁻¹

.

^(3.20)

From (3.6a) and (3.18), (3.20) becomes

H_k1

[

^Z3

(

^I

⊕

⁰m+^r

) +

^Z1

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)] +

^Hk2^∗

[

^Z4

(

^I

⊕

⁰m+^r

) +

^Z2

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)]

= −

^K01

[

^Z3

(

^I

⊕

^2Im

⊕

⁰r

) +

^Z1

(

⁰,˜

⊕

^Ir

) +

^O

(

k

)]

+

^K02^∗

[

^Z4

(

^I

⊕

^2Im

⊕

⁰r

) +

^Z2

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)].

^(3.21) Substituting (3.19) into (3.21) we get

H^∗_k2

(

Φ

+

^O

(

k

))[

^Z3

(

^I

⊕

⁰m+^r

) +

^Z1

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)] −

^Z4

(

^I

⊕

⁰m+^r

)

−

^Z2

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)

=

^O

(

¹

).

SinceΦ^Z1,b

=

^Z2,b, it holds that

H^∗_k2

([

Φ^Z3,a

−

^Z4,a

] +

^O

(

k

)) =

^O

(

¹

) ∈

C^n˜×

.

^(3.22) On the other hand, from (3.6a), (3.5) and (3.7), we have

A_k1Y₁

+

^Ak3Y₂

=

^A∗k1Y₁

J^∗₀2^k

⊕

Ω0^∗

2^k

+

^A∗k2Y₂

J₀^∗2^k

⊕ (

Ω0^∗

)

^{2k, (3.23)} A_k2Y₁

+

^Ak4Y₂

=

^A∗k3Y₁

J^∗₀2^k

⊕

Ω0^∗

2^k

+

^A∗k4Y₂ J^∗₀2^k

⊕

Ω0^∗

2^k

, (3.24)

A_k1Y₃

J²₀^k

⊕

^Ir

+

^Ak3Y₄

J₀^2k

⊕

^Ir

=

^A∗k1Y₃

+

^A∗k2Y₄ I_˜

⊕ (

Ω0^∗

)

^2k

+

^A∗k1Y₁

+

^A∗k2Y₂ 0_,˜

⊕

Γk^∗

, (3.25)

A_k2Y₃

J²₀^k

⊕

^Ir

+

^Ak4Y₄

J₀^2k

⊕

^Ir

=

^A∗k3Y₃

+

^A∗k4Y₄ I_˜

⊕ (

Ω0^∗

)

^2k

+

^A∗k3Y₁

+

^A∗k4Y₂ 0_,˜

⊕

Γk^∗

.

^(3.26)

As in (3.15) and (3.17), post-multiplying (3.25) by 0_˜,

⊕

Γk^∗

₋1 Ω0^∗

2^k

and substituting it into (3.23), we have

A_k1Y₁

+

^Ak3Y₂

=

^A∗k1Y₁

+

^A∗k2Y₂ J₀^∗2^k

⊕

⁰r

+

^A∗k1Y₃

+

^A∗k3Y₄ 0_˜,

⊕

^2−kΩ0^∗

−

^A∗k1Y3

+

^A∗k2Y4 0_˜,

⊕

^2−k

(

Ω0^∗

)

^2k+1

.

^(3.27) From (3.6a) and (3.18), (3.27) becomes

H_k1

Y₁

I_n

−

^J0^∗

2^k

⊕

⁰r

−

^Y3

0_˜,

⊕

^2−kΩ0^∗

I_r

−

Ω0^∗

2^k

+

^Hk2^∗

Y₂

I_n

−

^J∗0

2^k

⊕

⁰r

−

^Y4

0_˜,

⊕

^2−kΩ0^∗

I_r

−

Ω0^∗

2^k

= −

^K01

Y₁

I_n

+

^J0^∗

2^k

⊕

⁰r

−

^Y3

0_˜,

⊕

^2−kΩ0^∗

(

^Ir

+ (

Ω0^∗

)

^2k

)

+

^K02^∗

Y2

In

+

^J0^∗

2^k

⊕

^0r

−

^Y40_˜,

⊕

^2−kΩ0^∗

Ir

+ (

Ω0^∗

)

^{2k, (3.28)} and then

H_k1

= −

^H∗k2

(

Ψ

+

^O

(

k

)) −

^K01

+

^K02^∗Ψ

+

^O

(

k

)

^, whereΨ

=

^Y2Y₁⁻¹.

Post-multiplying (3.25) byI_˜

⊕

Γk^∗

₋1

and substituting (3.28) into it, we have H_k2^∗

Y4

(

^I

⊕

⁰m+^r

) +

^Y2

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

) − (

Ψ

+

^O

(

k

))[

^Y3

(

^I

⊕

⁰m+^r

) +

^Y1

(

⁰_,˜

⊕

^Ir

) +

^O

(

k

)]

=

^O

(

¹

).

SinceΨ^Y1,b

=

^Y2,b, it holds that

H_k2^∗

([

Ψ^Y3,a

−

^Y4,a

] +

^O

(

k

)) =

^O

(

¹

) ∈

C^n˜×

.

^(3.29) Combining (3.22) and (3.29) we get

H_k2^∗

([

Φ^Z3,a

−

^Z4,a

|

Ψ^Y3,a

−

^Y4,a

] +

^O

(

k

)) =

^O

(

¹

) ∈

C^n˜×2

.

^(3.30) By the assumption thatW

≡ [

Φ^Z3,a

−

^Z4,a

|

Ψ^Y3,a

−

^Y4,a

] ∈

C^n˜×2is of full row rank, it follows that H^∗_k2is uniformly bounded onk. Consequently, (3.19) implies that

^Hk1

, and in turn

^Ak1

^andA∗k2, are uniformly bounded onk. From (3.16), it follows that

A^∗_k1Z1

+

^A∗k2Z2

=

^O

(

k

) →

^{0, ask}

→ ∞.

^(3.31)

Applying the similar argument as in (3.15) and (3.17) to (3.24) and (3.26), we deduce that H_k4

= −

^Hk2

Y₂Y₁⁻¹

+

^O

(

k

)

+

^K04

−

^K02Y₂Y₁⁻¹

+

^O

(

k

).

Thus, (3.30) implies that

^Hk4

, and in turn

^Ak4

, are uniformly bounded onk.

To showA^∗_k3Z1

+

^A∗k4Z2

=

^O

(

k

)

, we post-multiply (3.14) by 0_˜,

⊕

Γk⁻¹Ω0^2kand obtain A^∗_k3Z₃

0_˜,

⊕

Γk⁻¹Ω0^2k

+

^A∗k4Z₄

0_˜,

⊕

Γk⁻¹Ω0^2k

= (

^Ak2Z₁

+

^Ak4Z₂

)

⁰

⊕

Ω0^2k

+ (

^Ak2Z₃

+

^Ak4Z₄

)

⁰_˜,

⊕

^2−kΩ0^2k+1

.

^(3.32)