JJ II

(1)

volume 6, issue 3, article 63, 2005.

Received 26 August, 2004;

accepted 24 May, 2005.

Communicated by:C.-K. Li

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

ON INVERSES OF TRIANGULAR MATRICES WITH MONOTONE ENTRIES

KENNETH S. BERENHAUT AND PRESTON T. FLETCHER

Department of Mathematics Wake Forest University

Winston-Salem, NC 27106, USA.

EMail:berenhks@wfu.edu

URL:http://www.math.wfu.edu/Faculty/berenhaut.html EMail:fletpt1@wfu.edu

c

2000Victoria University ISSN (electronic): 1443-5756 166-04

(2)

On Inverses of Triangular Matrices with Monotone Entries

Kenneth S. Berenhaut and Preston T. Fletcher

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of23

J. Ineq. Pure and Appl. Math. 6(3) Art. 63, 2005

http://jipam.vu.edu.au

Abstract

This note employs recurrence techniques to obtain entry-wise optimal inequalities for inverses of triangular matrices whose entries satisfy some monotonicity constraints. The derived bounds are easily computable.

2000 Mathematics Subject Classification:15A09, 39A10, 26A48.

Key words: Explicit bounds, Triangular matrix, Matrix inverse, Monotone entries, Off- diagonal decay, Recurrence relations.

We are very thankful to the referees for comments and insights that substantially improved this manuscript.

The first author acknowledges financial support from a Sterge Faculty Fellowship and an Archie fund grant.

1. Introduction

Much work has been done in the recent past to understand off-diagonal decay properties of structured matrices and their inverses (cf. Benzi and Golub [1], Demko, Moss and Smith [4], Eijkhout and Polman [5], Jaffard [6], Nabben [7] and [8], Peluso and Politi [9], Robinson and Wathen [10], Strohmer [11], Vecchio [12] and the references therein).

This paper studies nonnegative triangular matrices with off-diagonal decay.

In particular, let

L_n=





 l_1,1 l_2,1 l_2,2 l_3,1 l_3,2 l_3,3

... ... ... . ..

l_n,1 l_n,2 l_n,3 · · · l_n,n







be an invertible lower triangular matrix, and

X_n=L⁻¹_n =





 x_1,1 x_2,1 x_2,2 x_3,1 x_3,2 x_3,3

... ... ... . ..

x_n,1 x_n,2 x_n,3 · · · x_n,n





 ,

be its inverse.

We are interested in obtaining bounds on the entries inXn under the row- wise monotonicity assumption

(1.1) 0≤li,1 ≤li,2 ≤ · · · ≤li,i−1 ≤li,i

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of23

for2≤i≤n.

As an added generalization, we will consider[l_i,j]satisfying

(1.2) 0≤ l_i,1

l_i,i ≤ l_i,2

l_i,i ≤ · · · ≤ li,i−1

l_i,i ≤κi−1, for some nondecreasing sequenceκ= (κ₁, κ₂, κ₃, . . .).

The paper proceeds as follows. Section 2 contains some recurrence-type lemmas, while the main result, Theorem 3.1, and its proof are contained in Section3. The paper closes with some illustrative examples.

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of23

2. Preliminary Lemmas

In establishing our main results, we will employ recurrence techniques. In particular, suppose{bi}and{αi,j}satisfy the linear recurrence

(2.1) b_i =

i−1

X

k=0

(−α_i,k)b_k, (1≤i≤n),

withb₀ = 1and

(2.2) 0≤αi,0 ≤αi,1 ≤αi,2 ≤ · · · ≤αi,i−1 ≤Ai, fori≥1.

We will employ the following lemma, which reduces the scope of consider- ation in bounding solutions to (2.1).

Lemma 2.1. Suppose that {b_i} and{α_i,j}satisfy (2.1) and (2.2). Then, there exists a sequence a₁, a₂, . . . , a_n, with 0 ≤ a_i ≤ i for 1 ≤ i ≤ n, such that

|b_n| ≤ |d_n|, where{d_i}satisfiesd₀ = 1, and for1≤i≤n,

(2.3) d_i =





 Pi−1

j=ai(−Ai)dj, ifai < i

0, otherwise

.

In proving Lemma2.1, we will refer to the following result on inner prod- ucts.

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of23

Lemma 2.2. Suppose that p = (p₁, . . . , p_n)⁰ and q = (q₁, . . . , q_n)⁰ are n- vectors with

(2.4) 0≥p₁ ≥p₂ ≥ · · · ≥p_n≥ −A.

Define

(2.5) p^∗_n(ν, A) = (

ν

z }| { 0,0, . . . ,0,

n−ν

z }| {

−A, . . . ,−A,−A) for0≤ν≤n. Then,

(2.6) min

0≤ν≤n{p^∗_n(ν, A)·q} ≤p·q≤ max

0≤ν≤n{p^∗_n(ν, A)·q}, wherep·qdenotes the standard dot productPn

i=1p_iq_i. Proof. Supposepis of the form

(2.7) (p₁, . . . , p_j,

e1

z }| {

−k, . . . ,−k,

e2

z }| {

−A, . . . ,−A),

with0 ≥p₁ ≥ p₂ ≥ · · · ≥ p_j >−k > −A,e₁ ≥ 1ande₂ ≥ 0. First, assume thatp·q>0, and considerS =Pe1+j

i=j+1q_i. IfS <0then, sincek < A, (2.8) (p₁, p₂, . . . , pj−1, p_j,

e1

z }| {

−A, . . . ,−A

e2

z }| {

−A, . . . ,−A)·q≥p·q.

Otherwise, since−k < p_j, (2.9) (p1, p2, . . . , pj−1, pj,

e1

z }| { pj, . . . , pj,

e2

z }| {

−A, . . . ,−A)·q ≥p·q.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of23

In either case, there is a vector of the form in (2.7) with strictly less distinct values, whose inner product with q is at least as large as p ·q. Inductively, there exists a vector of the form in (2.7) with e₂ +e₁ = n, with as large, or larger, inner product. Hence, we have reduced to the case where p = (

e1

z }| {

−k, . . . ,−k,

e2

z }| {

−A, . . . ,−A), where e₁ = 0 and e_n = 0 are permissible. If k = 0 or e₁ = 0, then p = p^∗_n(e₁, A). Otherwise, consider S = Pe1

i=1q_i. If S < 0, then

(2.10) p^∗_n(0, A)·q≥p·q.

IfS ≥0,

(2.11) p^∗_n(e₁, A)·q ≥p·q.

The result for the casep·q>0now follows from (2.10) and (2.11).

The case whenp·q≤0is handled similarly, and the lemma follows.

We now turn to a proof of Lemma2.1.

Proof of Lemma2.1. The proof, here, involves applying Lemma2.2 to succes- sively “scale” the rows of the coefficient matrix







−α_1,0 0 . . . 0

−α_2,0 −α_2,1 . .. 0 ... ... . .. ...

−α_n,0 −α_n,1 · · · −αn,n−1





 ,

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of23

while not decreasing the value of|b_n|at any step.

First, define the sequences

¯

αi = (−αi,0, . . . ,−αi,i−1) and b^k,j = (b_k, . . . , b_j),

for0≤k ≤j ≤n−1and1≤i≤n.

Now, note that applying Lemma2.2to the vectorsp = α¯_n andq =b^0,n−1 yields a vectorp^∗(νn, An)(as in (2.5)) such that either

(2.12) p^∗(ν_n, A_n)·b^0,n−1 ≥α¯_n·b^0,n−1 =b_n>0 or

(2.13) p^∗(ν_n, A_n)·b^0,n−1 ≤α¯_n·b^0,n−1 =b_n≤0

Hence, suppose that the entries of the k^th through n^th rows of the coefficient matrix are of the form in (2.5), and express b_n as a linear combination of b₁, b₂, . . . , b_ki.e.

bn=

k

X

i=1

C_i^kbi

=C_k^kbk+

k−1

X

i=1

C_i^kbi. (2.14)

Now, supposeC_k^k > 0. As before, applying Lemma2.2 to the vectorsp =α¯_k andq=b^0,k−1yields a vectorp^∗_k(ν_k, A_k), such that

(2.15) p^∗_k(νk, Ak)·b^0,k−1 ≥α¯k·b^0,k−1 =bk.

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of23

Similarly, ifC_k^k ≤0, we obtain a vectorp^∗_k(ν_k, A_k), such that (2.16) p^∗_k(ν_k, A_k)·b^0,k−1 ≤α¯_k·b^0,k−1 =b_k.

Using the respective entries in p^∗_k(ν_k, A_k)in place of those in α¯_k in (2.1) will not decrease the value ofb_n. This completes the induction for the caseb_n > 0;

the caseb_n≤0is similar, and the lemma follows.

Remark 1. A version of Lemma2.3forA_i ≡1was recently applied in proving that all symmetric Toeplitz matrices generated by monotone convex sequences have off-diagonal decay preserved through triangular decompositions (see [2]).

Now, Fora= (A1, A2, A3, . . .), with

(2.17) 0≤A₁ ≤A₂ ≤A₃ ≤ · · · define

(2.18) Z_i(a)^def= max ( _i

Y

v=j

A_v : 1≤j ≤i )

,

fori≥1.

We have the following result on bounds for linear recurrences.

Lemma 2.3. Suppose that a = (A_j) satisfies the monotonicity constraint in (2.17). Then, fori≥1,

(2.19) sup{|b_i|:{b_j}and{α_i,j}satisfy (2.1) and (2.2)}=Z_i(a).

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of23

Proof. Suppose that{b_i}satisfies (2.1) and (2.2), and setζ_i =Z_i(a)andM_i = max{1, ζi}, fori≥1. From (2.18), we have

(2.20) A_i+1M_i =ζ_i+1,

fori≥1. By Lemma2.1, we may find sequences{d_i}and{a_i}satisfying (2.3) such that

(2.21) |d_n| ≥ |b_n|.

We will show that{di}satisfies the inequality

(2.22) |d_l+d_l+1+· · ·+d_i| ≤M_i, for0≤l ≤i.

Note that (2.22) (fori =n−1) and (2.3) imply thatd_n = 0ora_n ≤ n−1 and

|d_n|=

n−1

X

j=an

(−A_n)d_j

=A_n

n−1

X

j=an

d_j

≤A_nMn−1

=ζ_n. (2.23)

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of23

Sinced₀ = 1,d₁ ∈ {0,−A₁}and

max{|d₁|,|d₀+d₁|}= max{1, A₁,|1−A₁|}

= max{1, A₁}

=M1, (2.24)

i.e. the inequality in (2.22) holds fori = 1. Hence, suppose that (2.22) holds fori < N. RewritingdN, withv =aN, we have for0≤x≤N −1,

d_x+d_x+1+· · ·+d_N

= (d_x+d_x+1+· · ·+dN−1)−A_n(d_v +· · ·+dN−1)

=







(1−A_N)(d_v+· · ·+d_N−1) + (d_x+· · ·+dv−1), ifv > x (1−A_N)(d_x+· · ·+d_N−1)

−A_N(d_v+· · ·+dx−1), ifv ≤x . (2.25)

Let

S₁ =

( d_v+· · ·+d_N₋₁, ifv > x d_x+· · ·+dN−1, ifv ≤x

,

and

S2 =

( dx+· · ·+dv−1, ifv > x d_v+· · ·+dx−1, ifv ≤x .

In showing that|d_x+d_x+1+· · ·+d_N| ≤M_N, we will consider several cases depending on whetherA_N >1orA_N ≤1, and the signs ofS₁ andS₂.

Case 1 (AN >1andS1S2 >0)

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of23

1. v > x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁+S₂|

≤max{AN|S1|, AN|S2|}

≤A_Nmax{MN−1, Mv−1}

≤A_NMN−1

=ζ_N

=M_N, (2.26)

where the first inequality follows since(1−A_N)S₁andS₂are of opposite signs andAn >1. The second inequality follows from induction. The last equalities are direct consequences of the definition ofM_N and the fact that A_N > 1. The monotonicity of {M_i} is employed in obtaining the third inequality.

2. v ≤x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁−A_NS₂|

≤ |A_NS₁+A_NS₂|

=A_N|S₁+S₂|

=A_N|d_v+d_v+1+· · ·+dN−1|

≤A_NM_N−1

=ζN

=M_N. (2.27)

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of23

In (2.27), the first inequality follows since(1−A_N)S₁and−A_NS₂are of the same sign.

Case 2 (A_N >1andS₁S₂ ≤0) 1. v > x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁+S₂|

=| −A_NS₁+ (S₁+S₂)|.

(2.28)

IfS₁ andS₁+S₂ are of the same sign, then

| −A_NS₁+ (S₁+S₂)| ≤max{A_N|S₁|,|S₁+S₂|}

≤A_NM_N−1

=M_N. (2.29)

Otherwise,

| −A_NS₁+ (S₁+S₂)| ≤ | −A_NS₁+A_N(S₁ +S₂)|

=AN|S2|

≤A_NM_N−1

=M_N. (2.30)

2. v ≤x.

|dx+dx+1+· · ·+dN|=|(1−AN)S1−ANS2|

≤max{A_N|S₁|, A_N|S₂|}

≤A_NMN−1

=M_N (2.31)

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of23

Case 3 (A_N ≤1andS₁S₂ >0)

Note that forA_N ≤1,M_i = 1for alli.

1. v > x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁+S₂|

≤ |S1+S2|

≤MN−1

=M_N. (2.32)

2. v ≤x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁−A_NS₂|

≤max{|S₁|,|S₂|}

≤MN−1

=M_N. (2.33)

Case 4 (A_N ≤1andS₁S₂ ≤0) 1. v > x.

|dx+dx+1+· · ·+dN|=|(1−AN)S1+S2|

≤max{|S₁|,|S₂|}

≤max{MN−1, Mv−1}

=M_N. (2.34)

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of23

2. v ≤x.

|d_x+d_x+1+· · ·+d_N|=|(1−A_N)S₁−A_NS₂|

≤ |S1+S2|

≤MN−1

=M_N. (2.35)

α_i,j =











−A_h, ifi=h

−A_i, ifi > h,j =i 0, otherwise

. (2.36)

We close this section with an elementary result (without proof) which will serve to connect entries inL⁻¹_n with solutions to (2.1).

Lemma 2.4. SupposeM = [m_i,j]n×nandy= [y_i]n×1, satisfyM y = (1,0, . . . ,0)⁰, withM an invertible lower triangular matrix. Then,y1 = 1/m1,1, and

(2.37) y_i =

i−1

X

j=1

−m_i,j m_i,i

y_j,

for2≤i≤n.

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of23

3. The Main Result

We are now in a position to prove our main result.

Theorem 3.1. Supposeκ= (κ_i)satisfies

(3.1) 0≤κ₁ ≤κ₂ ≤κ₃ ≤ · · · , and set

(3.2) S ^def= {i:κ_i >1}.

As well, define{W_i,j}by

(3.3) Wi,j

def= Y

v∈(ST

{j,j+1,...,i−2}) S {i−1}

κv.

Then, for1≤i≤n,|x_i,i| ≤1/l_i,i and for1≤j < i≤n,

(3.4) |xi,j| ≤ W_i,j

l_j,j .

Proof. Suppose thatn≥1andX_n=L⁻¹_n . Solving for the sub-diagonal entries in thep^thcolumn ofX_nleads to the matrix equation





 lp,p

l_p+1,p l_p+1,p+1 ... ... . ..

l_n,p l_n,p+1 · · · l_n,n











 xp,p

x_p+1,p ... x_n,p







=





 1 0 ... 0





 .

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of23

Applying Lemma2.4givesx_p,p = 1/l_p,p, and

(3.5) x_p+i,p =

i−1

X

j=0

−l_p+i,p+j l_p+i,p+i

x_p+j,p,

for1≤i≤n−p.

Now, note that (1.2) gives (3.6) 0≤ l_p+i,p

l_p+i,p+i ≤ l_p+i,p+1

l_p+i,p+i ≤ · · · ≤ lp+i,p+i−1

l_p+i,p+i ≤κp+i−1. Hence by Lemma2.3,

|x_p+i,p| ≤ |x_p,p|Z_i((κ_p, κ_p+1, . . . , κ_p+i−1))

= 1

l_p,pWp+i,p, (3.7)

for1≤i≤n−p, and the theorem follows.

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of23

4. Examples

In this section, we provide examples to illustrate some of the structural infor- mation contained in Theorem3.1.

Example 4.1 (Equally spaced A_i). Suppose that A_i = Ci for i ≥ 1, where C > 0. Then, forn≥1,

Z_n(a) =











nC, C∈ 0,_n−1¹

; (n)_kC^k, C ∈ _n−k+1¹ ,_n−k¹

, (2≤k ≤n−1);

n!Cⁿ, C∈(1,∞), where(n)_k =n(n−1)· · ·(n−k+ 1).

Consider the matrix

L₇ =







1 0 0 0 0 0 0

0.25 1 0 0 0 0 0

0.5 0.5 1 0 0 0 0

0.75 0.75 0.75 1 0 0 0

1 1 1 1 1 0 0

0 1.25 1.25 1.25 1.25 1 0 1.5 1.5 1.5 1.5 1.5 1.5 1





 ,

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of23

with (rounded to three decimal places) X₇ =L⁻¹₇

=







1 0 0 0 0 0 0

−0.25 1 0 0 0 0 0

−0.375 −0.5 1 0 0 0 0

−0.281 −0.375 −0.75 1 0 0 0

−0.094 −0.125 −0.25 −1 1 0 0

1.25 0 0 0 −1.25 1 0

−1.875 0 0 0 0.375 −1.5 1





 . (4.1)

Applying Theorem 3.1, withκ = (.25, .50, .75,1.00,1.25,1.50, . . .) gives the entry-wise bounds

(4.2)







1 0 0 0 0 0 0

0.25 1 0 0 0 0 0

0.5 0.5 1 0 0 0 0

0.75 0.75 0.75 1 0 0 0

1 1 1 1 1 0 0

1.25 1.25 1.25 1.25 1.25 1 0 1.875 1.875 1.875 1.875 1.875 1.5 1





 .

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of23

Comparing (4.1) and (4.2), the absolute values of entry-wise ratios are

(4.3)





 1

1 1

0.75 1 1 0.375 0.5 1 1 0.094 0.125 0.25 1 1

1 0 0 0 1 1

1 0 0 0 0.2 1 1





 .

Note that here L₇ was constructed so that|x_7,1| = W_7,1. In fact, as suggested by (2.19), for each4-tuple(κ, I, J, n)with1 ≤J ≤ I ≤n, there exists a pair (L_n,X_n)satisfying (1.2) withX_n = (x_i,j) = L⁻¹_n , such that|x_I,J|=W_I,J. Example 4.2 (Constant Ai). Suppose that Ai = C for i ≥ 1, where C > 0.

Then, forn≥1,

Z_n(a) =

( C, ifC ≤1 Cⁿ, ifC > 1

.

In [3], the following theorem was obtained when (2.2) is replaced with

(4.4) 0≤α_i,j ≤A,

for0≤j ≤i−1andi≥1.

Theorem 4.1. Suppose that A > 0 and m = [1/A], where square brackets indicate the greatest integer function. If{Λ_j}^∞_j=1is defined by

(4.5) Λn = max{|bn|:{bi}and[αi,j]satisfy (2.1) and (4.4)},

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of23

forn≥1, then

(4.6) Λn=











A, ifn = 1

max(A, A²), ifn = 2 _n−2

2

n−1 2

A³+A, if3≤n ≤2m+ 1 (n−2)A², ifn = 2m+ 2 AΛn−1+ Λn−2, ifn ≥2m+ 3

.

Proof. See [3].

Thus, if the monotonicity assumption in (2.2) is dropped the scenario is much different. In fact, in (4.6),{Λn}increases at an exponential rate for allA >0.

This leads to the following question.

Open Question. Set

Λ^∗_n = max{|b_n|:{b_i}and[α_i,j]

satisfy (2.1) andα_i,j ≤A_ifor0≤j ≤i−1}.

(4.7)

What is the value ofΛ^∗_nin terms of the sequence{A_i}and its assorted properties (eg. monotonicity, convexity etc.)?

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of23

References

[1] M. BENZI, AND G. GOLUB, Bounds for the entries of matrix functions with applications to preconditioning, BIT, 39(3) (1999), 417–438.

[2] K.S. BERENHAUT ANDD. BANDYOPADHYAY, Monotone convex se- quences and Cholesky decomposition of symmetric Toeplitz matrices, Lin- ear Algebra and Its Applications, 403 (2005), 75–85.

[3] K.S. BERENHAUTANDD.C. MORTON, Second order bounds for linear recurrences with negative coefficients, in press, J. of Comput. and App.

Math., (2005).

[4] S. DEMKO, W. MOSS, ANDP. SMITH, Decay rates for inverses of band matrices, Math. Comp., 43 (1984), 491–499.

[5] V. EIJKHOUT AND B. POLMAN, Decay rates of inverses of bandedm- matrices that are near to Toeplitz matrices, Linear Algebra Appl., 109 (1988), 247–277.

[6] S. JAFFARD, Propriétés des matrices “bien localisées" près de leur diag- onale et quelques applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(5) (1990), 461–476.

[7] R. NABBEN, Decay rates of the inverse of nonsymmetric tridiagonal and band matrices, SIAM J. Matrix Anal. Appl., 20(3) (1999), 820–837.

[8] R. NABBEN, Two-sided bounds on the inverses of diagonally dominant tridiagonal matrices, Special issue celebrating the 60th birthday of Ludwig Elsner, Linear Algebra Appl., 287(1-3) (1999), 289–305.

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of23

[9] R. PELUSO, ANDT. POLITI, Some improvements for two-sided bounds on the inverse of diagonally dominant tridiagonal matrices, Linear Algebra Appl., 330(1-3) (2001), 1–14.

[10] P.D. ROBINSON AND A.J. WATHEN, Variational bounds on the entries of the inverse of a matrix, IMA J. Numer. Anal., 12(4) (1992), 463–486.

[11] T. STROHMER, Four short stories about Toeplitz matrix calculations, Lin- ear Algebra Appl., 343/344 (2002), 321–344.

[12] A. VECCHIO, A bound for the inverse of a lower triangular Toeplitz ma- trix, SIAM J. Matrix Anal. Appl., 24(4) (2003), 1167–1174.