Journal of Inequalities in Pure and
Applied Mathematics
ON INVERSES OF TRIANGULAR MATRICES WITH
MONOTONE ENTRIES
volume 6, issue 3, article 63,
2005.
KENNETH S. BERENHAUT AND PRESTON T. FLETCHER
Department of Mathematics
Wake Forest University
Winston-Salem, NC 27106, USA.
Received 26 August, 2004;
accepted 24 May, 2005.
Communicated by: C.-K. Li
EMail: berenhks@wfu.edu
URL: http://www.math.wfu.edu/Faculty/berenhaut.html
EMail: fletpt1@wfu.edu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
166-04
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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, Offdiagonal 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.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
References
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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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


l1,1
 l2,1 l2,2





Ln =  l3,1 l3,2 l3,3

 ..

..
..
.
.
 .

.
.
.
ln,1 ln,2 ln,3 · · · ln,n
be an invertible lower triangular matrix, and

x1,1
 x2,1 x2,2

 x3,1 x3,2 x3,3
=
X n = L−1

n
 ..
..
..
...
 .
.
.
xn,1 xn,2 xn,3 · · · xn,n
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
Title Page
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



,


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be its inverse.
We are interested in obtaining bounds on the entries in X n under the rowwise monotonicity assumption
Page 3 of 23
0 ≤ li,1 ≤ li,2 ≤ · · · ≤ li,i−1 ≤ li,i
J. Ineq. Pure and Appl. Math. 6(3) Art. 63, 2005
(1.1)
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for 2 ≤ i ≤ n.
As an added generalization, we will consider [li,j ] satisfying
(1.2)
0≤
li,1
li,2
li,i−1
≤
≤ ··· ≤
≤ κi−1 ,
li,i
li,i
li,i
for some nondecreasing sequence κ = (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
Section 3. The paper closes with some illustrative examples.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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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)
bi =
i−1
X
(−αi,k )bk , (1 ≤ i ≤ n),
k=0
with b0 = 1 and
(2.2)
On Inverses of Triangular
Matrices with Monotone Entries
0 ≤ αi,0 ≤ αi,1 ≤ αi,2 ≤ · · · ≤ αi,i−1 ≤ Ai ,
for i ≥ 1.
We will employ the following lemma, which reduces the scope of consideration in bounding solutions to (2.1).
Lemma 2.1. Suppose that {bi } and {αi,j } satisfy (2.1) and (2.2). Then, there
exists a sequence a1 , a2 , . . . , an , with 0 ≤ ai ≤ i for 1 ≤ i ≤ n, such that
|bn | ≤ |dn |, where {di } satisfies d0 = 1, and for 1 ≤ i ≤ n,
 Pi−1
 j=ai (−Ai )dj , if ai < i
(2.3)
di =
.

0,
otherwise
In proving Lemma 2.1, we will refer to the following result on inner products.
Kenneth S. Berenhaut and
Preston T. Fletcher
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Lemma 2.2. Suppose that p = (p1 , . . . , pn )0 and q = (q1 , . . . , qn )0 are nvectors with
0 ≥ p1 ≥ p2 ≥ · · · ≥ pn ≥ −A.
(2.4)
Define
n−ν
ν
(2.5)
p∗n (ν, A)
}|
{
z }| { z
= (0, 0, . . . , 0, −A, . . . , −A, −A)
On Inverses of Triangular
Matrices with Monotone Entries
for 0 ≤ ν ≤ n. Then,
(2.6)
min {p∗n (ν, A) · q} ≤ p · q ≤ max {p∗n (ν, A) · q},
0≤ν≤n
0≤ν≤n
where p · q denotes the standard dot product
Pn
i=1
p i qi .
Title Page
Proof. Suppose p is of the form
e
(2.7)
e
z
}|1
{ z
}|2
{
(p1 , . . . , pj , −k, . . . , −k, −A, . . . , −A),
with 0 ≥ p1 ≥ p2 ≥ · · · ≥ pj > P
−k > −A, e1 ≥ 1 and e2 ≥ 0. First, assume
1 +j
that p · q > 0, and consider S = ei=j+1
qi . If S < 0 then, since k < A,
e
(2.8)
Kenneth S. Berenhaut and
Preston T. Fletcher
e
z
}|1
{z
}|2
{
(p1 , p2 , . . . , pj−1 , pj , −A, . . . , −A −A, . . . , −A) · q ≥ p · q.
Otherwise, since −k < pj ,
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e
(2.9)
e
}|2
{
z }|1 { z
(p1 , p2 , . . . , pj−1 , pj , pj , . . . , pj , −A, . . . , −A) · q ≥ p · q.
J. Ineq. Pure and Appl. Math. 6(3) Art. 63, 2005
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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 =
e
e
z
}|1
{ z
}|2
{
(−k, . . . , −k, −A, . . . , −A), where e1 = 0 and en = 0 are permissible.
If
Pe1
∗
k = 0 or e1 = 0, then p = pn (e1 , A). Otherwise, consider S = i=1 qi . If
S < 0, then
(2.10)
p∗n (0, A) · q ≥ p · q.
If S ≥ 0,
(2.11)
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
p∗n (e1 , A) · q ≥ p · q.
The result for the case p · q > 0 now follows from (2.10) and (2.11).
The case when p · q ≤ 0 is handled similarly, and the lemma follows.
We now turn to a proof of Lemma 2.1.
Proof of Lemma 2.1. The proof, here, involves applying Lemma 2.2 to successively “scale” the rows of the coefficient matrix


−α1,0
0
...
0




..
.
0
 −α2,0 −α2,1


,


..
..
..
..


.
.
.
.


−αn,0 −αn,1 · · · −αn,n−1
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while not decreasing the value of |bn | at any step.
First, define the sequences
ᾱi = (−αi,0 , . . . , −αi,i−1 ) and
bk,j = (bk , . . . , bj ),
for 0 ≤ k ≤ j ≤ n − 1 and 1 ≤ i ≤ n.
Now, note that applying Lemma 2.2 to the vectors p = ᾱn and q = b0,n−1
yields a vector p∗ (νn , An ) (as in (2.5)) such that either
p∗ (νn , An ) · b0,n−1 ≥ ᾱn · b0,n−1 = bn > 0
(2.12)
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
or
p∗ (νn , An ) · b0,n−1 ≤ ᾱn · b0,n−1 = bn ≤ 0
(2.13)
Hence, suppose that the entries of the k th through nth rows of the coefficient
matrix are of the form in (2.5), and express bn as a linear combination of
b1 , b2 , . . . , bk i.e.
bn =
k
X
Cik bi
i=1
= Ckk bk +
(2.14)
Cik bi .
suppose Ckk
0,k−1
(2.15)
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k−1
X
i=1
Now,
and q = b
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> 0. As before, applying Lemma 2.2 to the vectors p = ᾱk
yields a vector p∗k (νk , Ak ), such that
p∗k (νk , Ak ) · b0,k−1 ≥ ᾱk · b0,k−1 = bk .
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Similarly, if Ckk ≤ 0, we obtain a vector p∗k (νk , Ak ), such that
(2.16)
p∗k (νk , Ak ) · b0,k−1 ≤ ᾱk · b0,k−1 = bk .
Using the respective entries in p∗k (νk , Ak ) in place of those in ᾱk in (2.1) will
not decrease the value of bn . This completes the induction for the case bn > 0;
the case bn ≤ 0 is similar, and the lemma follows.
Remark 1. A version of Lemma 2.3 for Ai ≡ 1 was recently applied in proving
that all symmetric Toeplitz matrices generated by monotone convex sequences
have off-diagonal decay preserved through triangular decompositions (see [2]).
Kenneth S. Berenhaut and
Preston T. Fletcher
Now, For a = (A1 , A2 , A3 , . . . ), with
(2.17)
0 ≤ A1 ≤ A2 ≤ A3 ≤ · · ·
Title Page
define
Contents
(
(2.18)
def
Zi (a) = max
i
Y
)
Av : 1 ≤ j ≤ i
,
v=j
for i ≥ 1.
We have the following result on bounds for linear recurrences.
Lemma 2.3. Suppose that a = (Aj ) satisfies the monotonicity constraint in
(2.17). Then, for i ≥ 1,
(2.19)
On Inverses of Triangular
Matrices with Monotone Entries
sup{|bi | : {bj } and {αi,j } satisfy (2.1) and (2.2)} = Zi (a).
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Proof. Suppose that {bi } satisfies (2.1) and (2.2), and set ζi = Zi (a) and Mi =
max{1, ζi }, for i ≥ 1. From (2.18), we have
(2.20)
Ai+1 Mi = ζi+1 ,
for i ≥ 1. By Lemma 2.1, we may find sequences {di } and {ai } satisfying (2.3)
such that
(2.21)
|dn | ≥ |bn |.
We will show that {di } satisfies the inequality
(2.22)
|dl + dl+1 + · · · + di | ≤ Mi ,
for 0 ≤ l ≤ i.
Note that (2.22) (for i = n − 1) and (2.3) imply that dn = 0 or an ≤ n − 1
and
n−1
X
|dn | = (−An )dj j=an
n−1
X
= An dj j=an
(2.23)
≤ An Mn−1
= ζn .
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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Since d0 = 1, d1 ∈ {0, −A1 } and
(2.24)
max{|d1 |, |d0 + d1 |} = max{1, A1 , |1 − A1 |}
= max{1, A1 }
= M1 ,
i.e. the inequality in (2.22) holds for i = 1. Hence, suppose that (2.22) holds
for i < N . Rewriting dN , with v = aN , we have for 0 ≤ x ≤ N − 1,
dx + dx+1 + · · · + dN
= (dx + dx+1 + · · · + dN −1 ) − An (dv + · · · + dN −1 )


 (1 − AN )(dv + · · · + dN −1 ) + (dx + · · · + dv−1 ), if v > x
(2.25) =
.
(1 − AN )(dx + · · · + dN −1 )


−AN (dv + · · · + dx−1 ),
if v ≤ x
Let
(
S1 =
dv + · · · + dN −1 , if v > x
dx + · · · + dN −1 , if v ≤ x
,
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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and
(
S2 =
dx + · · · + dv−1 , if v > x
dv + · · · + dx−1 , if v ≤ x
.
In showing that |dx + dx+1 + · · · + dN | ≤ MN , we will consider several cases
depending on whether AN > 1 or AN ≤ 1, and the signs of S1 and S2 .
Case 1 (AN > 1 and S1 S2 > 0)
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1. v > x.
(2.26)
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 + S2 |
≤ max{AN |S1 |, AN |S2 |}
≤ AN max{MN −1 , Mv−1 }
≤ AN MN −1
= ζN
= MN ,
where the first inequality follows since (1 − AN )S1 and S2 are of opposite
signs and An > 1. The second inequality follows from induction. The last
equalities are direct consequences of the definition of MN and the fact that
AN > 1. The monotonicity of {Mi } is employed in obtaining the third
inequality.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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2. v ≤ x.
(2.27)
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 − AN S2 |
≤ |AN S1 + AN S2 |
= AN |S1 + S2 |
= AN |dv + dv+1 + · · · + dN −1 |
≤ AN MN −1
= ζN
= MN .
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In (2.27), the first inequality follows since (1 − AN )S1 and −AN S2 are of
the same sign.
Case 2 (AN > 1 and S1 S2 ≤ 0)
1. v > x.
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 + S2 |
= | − AN S1 + (S1 + S2 )|.
(2.28)
If S1 and S1 + S2 are of the same sign, then
| − AN S1 + (S1 + S2 )| ≤ max{AN |S1 |, |S1 + S2 |}
≤ AN MN −1
= MN .
(2.29)
| − AN S1 + (S1 + S2 )| ≤ | − AN S1 + AN (S1 + S2 )|
= AN |S2 |
≤ AN MN −1
= MN .
2. v ≤ x.
(2.31)
Kenneth S. Berenhaut and
Preston T. Fletcher
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Otherwise,
(2.30)
On Inverses of Triangular
Matrices with Monotone Entries
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|dx + dx+1 + · · · + dN | = |(1 − AN )S1 − AN S2 |
≤ max{AN |S1 |, AN |S2 |}
≤ AN MN −1
= MN
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Case 3 (AN ≤ 1 and S1 S2 > 0)
Note that for AN ≤ 1, Mi = 1 for all i.
1. v > x.
(2.32)
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 + S2 |
≤ |S1 + S2 |
≤ MN −1
= MN .
Kenneth S. Berenhaut and
Preston T. Fletcher
2. v ≤ x.
(2.33)
On Inverses of Triangular
Matrices with Monotone Entries
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 − AN S2 |
≤ max{|S1 |, |S2 |}
≤ MN −1
= MN .
Case 4 (AN ≤ 1 and S1 S2 ≤ 0)
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1. v > x.
Close
(2.34)
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 + S2 |
≤ max{|S1 |, |S2 |}
≤ max{MN −1 , Mv−1 }
= MN .
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2. v ≤ x.
(2.35)
|dx + dx+1 + · · · + dN | = |(1 − AN )S1 − AN S2 |
≤ |S1 + S2 |
≤ MN −1
= MN .
Thus, in all cases |dx + dx+1 + · · · + dN | ≤ MN and hence by (2.23), |dN | ≤
ζN . Equation (2.19) now follows since, for 1 ≤ h ≤ n, |bn | = Ah Ah+1 · · · An
is attained for [αi,j ] defined by

−Ah , if i = h



−Ai , if i > h, j = i .
(2.36)
αi,j =



0,
otherwise
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
Title Page
Contents
We close this section with an elementary result (without proof) which will
serve to connect entries in L−1
n with solutions to (2.1).
Lemma 2.4. Suppose M = [mi,j ]n×n and y = [yi ]n×1 , satisfy M y = (1, 0, . . . , 0)0 ,
with M an invertible lower triangular matrix. Then, y1 = 1/m1,1 , and
i−1 X
mi,j
(2.37)
yi =
−
yj ,
mi,i
j=1
for 2 ≤ i ≤ n.
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3.
The Main Result
We are now in a position to prove our main result.
Theorem 3.1. Suppose κ = (κi ) satisfies
0 ≤ κ1 ≤ κ2 ≤ κ3 ≤ · · · ,
(3.1)
and set
def
On Inverses of Triangular
Matrices with Monotone Entries
S = {i : κi > 1}.
(3.2)
As well, define {Wi,j } by
(3.3)
Kenneth S. Berenhaut and
Preston T. Fletcher
def
Y
Wi,j =
v ∈ (S
T
{j,j+1,...,i−2})
κv .
S
{i−1}
Then, for 1 ≤ i ≤ n, |xi,i | ≤ 1/li,i and for 1 ≤ j < i ≤ n,
(3.4)
|xi,j | ≤
Wi,j
.
lj,j
Proof. Suppose that n ≥ 1 and X n = L−1
n . Solving for the sub-diagonal entries
th
in the p column of X n leads to the matrix equation


  
xp,p
1
lp,p
 lp+1,p lp+1,p+1
  xp+1,p   0 


  
 ..


 =  . .
..
.
.
..
 .
  ..   .. 
.
ln,p
ln,p+1 · · · ln,n
xn,p
0
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Applying Lemma 2.4 gives xp,p = 1/lp,p , and
i−1 X
lp+i,p+j
=
−
xp+j,p ,
l
p+i,p+i
j=0
xp+i,p
(3.5)
for 1 ≤ i ≤ n − p.
Now, note that (1.2) gives
(3.6)
0≤
lp+i,p
lp+i,p+i
≤
lp+i,p+1
lp+i,p+i−1
≤ ··· ≤
≤ κp+i−1 .
lp+i,p+i
lp+i,p+i
Hence by Lemma 2.3,
(3.7)
|xp+i,p | ≤ |xp,p |Zi ((κp , κp+1 , . . . , κp+i−1 ))
1
=
Wp+i,p ,
lp,p
for 1 ≤ i ≤ n − p, and the theorem follows.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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4.
Examples
In this section, we provide examples to illustrate some of the structural information contained in Theorem 3.1.
Example 4.1 (Equally spaced Ai ). Suppose that Ai = Ci for i ≥ 1, where
C > 0. Then, for n ≥ 1,

1
nC,
C
∈
0,
;

n−1


1
1
(n)k C k , C ∈ n−k+1
, n−k
, (2 ≤ k ≤ n − 1);
Zn (a) =



n!C n ,
C ∈ (1, ∞),
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
where (n)k = n(n − 1) · · · (n − k + 1).
Title Page
Consider the matrix

1
0
0
0
0
0
 0.25 1
0
0
0
0


 0.5 0.5
1
0
0
0

0
0
L7 = 
 0.75 0.75 0.75 1
 1
1
1
1
1
0


 0 1.25 1.25 1.25 1.25 1
1.5 1.5 1.5 1.5 1.5 1.5
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0
0
0
0
0
0
1






,





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with (rounded to three decimal places)
X7 = L−1
7
(4.1)





=





1
0
0
0
0
0
−0.25
1
0
0
0
0
−0.375 −0.5
1
0
0
0
−0.281 −0.375 −0.75 1
0
0
−0.094 −0.125 −0.25 −1
1
0
1.25
0
0
0 −1.25
1
−1.875
0
0
0 0.375 −1.5
Applying Theorem 3.1,
entry-wise bounds

1
 0.25


 0.5

 0.75
(4.2)

 1


 1.25
1.875
0
0
0
0
0
0
1






.





On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
with κ = (.25, .50, .75, 1.00, 1.25, 1.50, . . . ) gives the
Title Page
0
0
0
0
0
1
0
0
0
0
0.5
1
0
0
0
0.75 0.75
1
0
0
1
1
1
1
0
1.25 1.25 1.25 1.25 1
1.875 1.875 1.875 1.875 1.5
0
0
0
0
0
0
1






.





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Comparing (4.1) and (4.2), the absolute values of entry-wise ratios are


1

 1
1



 0.75
1
1


 0.375 0.5
.
1
1
(4.3)



 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 |x7,1 | = W7,1 . In fact, as suggested
by (2.19), for each 4-tuple (κ, I, J, n) with 1 ≤ J ≤ I ≤ n, there exists a pair
(Ln , X n ) satisfying (1.2) with X n = (xi,j ) = L−1
n , such that |xI,J | = WI,J .
Example 4.2 (Constant Ai ). Suppose that Ai = C for i ≥ 1, where C > 0.
Then, for n ≥ 1,
(
C,
if C ≤ 1
Zn (a) =
.
C n , if C > 1
In [3], the following theorem was obtained when (2.2) is replaced with
0 ≤ αi,j ≤ A,
(4.4)
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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for 0 ≤ j ≤ i − 1 and i ≥ 1.
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Theorem 4.1. Suppose that A > 0 and m = [1/A], where square brackets
indicate the greatest integer function. If {Λj }∞
j=1 is defined by
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Λn = max{|bn | : {bi } and [αi,j ] satisfy (2.1) and (4.4)},
J. Ineq. Pure and Appl. Math. 6(3) Art. 63, 2005
(4.5)
http://jipam.vu.edu.au
for n ≥ 1, then
(4.6)

A,






max(A, A2 ),


 n−1 3
n−2
A + A,
Λn =
2
2





(n − 2)A2 ,




AΛn−1 + Λn−2 ,
if n = 1
if n = 2
if 3 ≤ n ≤ 2m + 1 .
if n = 2m + 2
if n ≥ 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 all A > 0.
This leads to the following question.
Open Question. Set
(4.7)
Λ∗n = max{|bn | : {bi } and [αi,j ]
satisfy (2.1) and αi,j ≤ Ai for 0 ≤ j ≤ i − 1}.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
Title Page
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What is the value of Λ∗n in terms of the sequence {Ai } and its assorted properties
(eg. monotonicity, convexity etc.)?
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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 AND D. BANDYOPADHYAY, Monotone convex sequences and Cholesky decomposition of symmetric Toeplitz matrices, Linear Algebra and Its Applications, 403 (2005), 75–85.
[3] K.S. BERENHAUT AND D.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, AND P. 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 banded mmatrices that are near to Toeplitz matrices, Linear Algebra Appl., 109
(1988), 247–277.
[6] S. JAFFARD, Propriétés des matrices “bien localisées" près de leur diagonale 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.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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[9] R. PELUSO, AND T. 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, Linear Algebra Appl., 343/344 (2002), 321–344.
[12] A. VECCHIO, A bound for the inverse of a lower triangular Toeplitz matrix, SIAM J. Matrix Anal. Appl., 24(4) (2003), 1167–1174.
On Inverses of Triangular
Matrices with Monotone Entries
Kenneth S. Berenhaut and
Preston T. Fletcher
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