Advances in Linear Algebra & Matrix Theory, 2013, 3, 17-21
http://dx.doi.org/10.4236/alamt.2013.33004 Published Online September 2013 (http://www.scirp.org/journal/alamt)
Innovative Structured Matrices
Rahul Gupta, Garimella Rama Murthy
IIIT-Hyderabad,Gachibowli, Hyderabad, India
Received February 17, 2013; revised March 17, 2013; accepted April 2, 2013
Copyright © 2013 Rahul Gupta, Garimella Rama Murthy. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
ABSTRACT
Various directions of obtaining novel structured matrices are discussed. A new class of matrices, called “the L-family”
matrices are introduced and their properties are studied.
Keywords: Innovative Matrices; L Matrices; Structured Matrices
1. Introduction
Linear algebra is central to modern mathematics and has
been found many applications in Science, Technology,
Engineering and many other disciplines. Matrices with
special kind of structure like Toeplitz, Hankel etc., are
studied with great interest [2]. In this paper, a special
family (called “The L-family”) of matrices are discussed
in detail with some interesting properties. It is expected
that this family of matrices will find interesting applica-
tions in various disciplines of human endeavour.
This idea of analysing new structured matrices was
adopted from Dr. G. Rama Murthy’s journal paper “In-
novative Structured Matrices”, International Journal of
Algorithms, Computing and Mathematics Volume 2,
Number 4, November 2009.
2. Logical Idea behind Structured Matrices
We can think of innovative structured matrices in many
ways. For example, one way is to construct a matrix from
the indices or subscripts of elements of the matrix. The
other way is to assign a particular same value to all ele-
ments for each subset of the matrix, where these subsets
are taken to be mutually exclusive and exhaustive [1,3].
Constructing matrices from indices point-of-view:
11 1213
21 2223
31 3233
aaa
aaa
aaa





we can map axy to a function of x, y, f (x, y).
x, y = 1, 2, 3 i.e.,


,
xy
afxy
The following is the matrix constructed by taking


22
,
f
xyxy xy
2612
61630
12 30 54





We can also take just like

,
xy
aafxy



,
f
xyx y

 as in a Toeplitz.
Constructing matrices from subset point-of-view:
Let us look at some typical examples.
Example: 1
33333
32223
32123
32223
33333
aaaaa
aaaaa
aaaaa
aaaaa
aaaaa








 constructed by taking size-increasing, mutually ex-
clusive and exhaustive square shaped subsets
Example: 2
4321
4322
4333
4444
aaaa
aaaa
aaaa
aaaa






1234
2234
3334
4444
aaaa
aaaa
aaaa
aaaa






4444
4333
4322
4321
aaaa
aaaa
aaaa
aaaa






4444
3334
2234
1234
aaaa
aaaa
aaaa
aaaa






The above 4 matrices are constructed by taking └, ┘,
C
opyright © 2013 SciRes. ALAMT

R. GUPTA, G. R. MURTHY 18
┐, ┌ shaped subsets of the matrix as the criteria respec-
tively.
Example: 3
2222
2112
2112
2112
aaaa
aaaa
aaaa
aaaa






2222
2111
2111
2222
aaaa
aaaa
aaaa
aaaa










2222
1112
1112
2222
aaaa
aaaa
aaaa
aaaa






2112
2112
2112
2222
aaaa
aaaa
aaaa
aaaa






These matrices are constructed by taking shaped
subsets of the matrix as the criteria in all the four direc-
tions with opening towards south, east, west and north
directions respectively.
Remark: This logical approach can be extended to ar-
rive at large number of structured matrices (like Toeplitz
matrix) [3].
The L-family of matrices:
Now we focus our attention on the class of matrices in
Example 2.
This class of matrices can be called “The L-family”
matrices due to their resemblance in their structure with
the letter “L”.
4321
4322
4333
4444
L-matrix
aaa a
aaaa
aaaa
aaaa






1234
2234
3334
4444
rev-L matrix
aaaa
aaaa
aaaa
aaaa






4444
4333
4322
4321
inv-L matrix
aaaa
aaaa
aaaa
aaa a






4444
3334
2234
1234
rev-inv-L matrix
aaaa
aaaa
aaaa
aaaa






[rev: stands for reverse and inv: stands for inverse].
Let us consider only square matrices in our entire dis-
cussion.
Let us define an originator of a matrix in L-family.
The ith originator of a n*n L-family matrix is the element
which occurs (2i-1) times in the matrix.
In all the above mentioned matrices, a1-1st originator,
a2-2nd originator, a3-3rd originator and a4-4th originator.
Let us examine some of the properties of L-family:
Claim 1: For any L-family matrix A, 1
A
A


where ||.|| represents natural norm [4].
Proof 1:
1
11
11
1
max max
maximum absolute column sum of the matrix
m
zij
zjn
i
AA a








11
1
max max
maximum absolute row sum of the matrix
n
zij
zjm
i
AA a










For rev-L and inv-L matrices absolute kth row sum is
equal to absolute kth column sum for . For
L- and rev-inv-L matrices absolute kth column sum is
equal to absolute (n ‒ k + 1)th column sum for
1, 2,,kn
1, 2,,kn

. (Note: m = n for a square matrix).
Hence 1
A
A


Claim 2: For an L-family matrix to be stochastic, all
the originators of it must be equal to each other.
Proof 2: The sum of all the elements in each column
of a stochastic matrix is equal to 1.Consider an L-matrix
of order “n”, i.e.,


,1,2,,
i
ai n are originators.
Sum of all the elements in 1st column = nan
11
nn
naa n 
Sum of all the elements in 2nd column =


1
1nn na a





11
111 1
nn
na na

 n
Sum of all the elements in 3rd column =


21
2nn naaa
n




22
221 1
nn
nan a

 n
S imilarly, sum of all the elements in kth column =




1
11
nk
nk akn

 


1
1
11
1for4,5,
nk
nk
nk akn
ank


 
 
1
n
Therefore, 1
i
an for all in
1, 2,,
Hence all the originators must be equal to each other.
Similar type proof can be provided for other type of
matrices rev-L, inv-L, rev-inv-L matrices also.
Claim 3: The determinant of a rev-L or inv-L matrix
with originators


,1,2,,
i
ai n is equal to





 
122334 1nn
aa aaa aaaa

 n
.
Proof 3: The proof for rev-L matrix is as follows.
Perform the following elementary row operations on
the determinant.
1) 1iii
RRR

 for all in

1, 2,,1
2) Take






12 231
,,aaa aaa

 ,,
n
ammon
out of the determinant
nn
co
3) 1iii RRR

 for all in

,1,,n2
4) The remaining determinant goes to “1” as it is Iden-
tity matrix.

R. GUPTA, G. R. MURTHY 19
Hence proved for rev-L matrix.
The proof for inv-L matrix is as follows.
1) for all
1iii
RRR



,1,,inn2
2) Take an, (a1 − a2), (a2 − a3), ... (an−1 − an) common
out of the determinant
3) for all
1iii
RRR



1, 2,,1in


4) The remaining determinant goes to “1” as it is Iden-
tity matrix.
Hence proved for inv-L matrix also.
Claim 4: The determinant of L-matrix or rev-inv-L
matrix with originators


,1,2,,
i
ai n is equal to


 
2
122334 1
1n
n
aa aaaaaaa



n n

 
,
where [.] denotes step function/greatest integer function.
The above claim can be easily proved using a simple
mathematical induction. Before going through the proof
let us look at some criteria which will be useful in prov-
ing the claim.
Let us define Mirror image of a n*n ordered square
matrix as the matrix , where
.
 
ij
Aa


 
1in j
ba

 
ij
Bb


 
ij
(MIRROR)
1112131312 11
21 2223232221
31 32 3333 3231
aaa aaa
aaaaaa
aaa aaa
 
 
 
 
 





001
010
100





(say M3) is the mirror image of Identity
matrix, I3.
Our aim is to find out the determinant (say Dn) of Mn.
Claim 5:


2
1n
n
D
Proof 5: We shall prove this using mathematical in-
duction method.
Let


12
11k k
D


 .
Then
 



11121
1
1111
kkkk
kk
DD  

 

 12k

Case 1: If k is even (= 2p)


 
 

211221 13
22
11
11 1
pppp p
k
pk k
D   
 
1
Case 2: If k is odd (= 2p + 1)


 



211122
111
pppp
k
D   1
k
.
Hence,


2
1n
n
D .
The proof for claim 4 is as follows.
Proof 4: The proof for L-matrix is as follows:
Perform the following elementary row operations on
the determinant.
5) 1iii
RRR

 for all

1, 2,,1in
6) Take






12 231
,,,,aaa aaa

 
n
ammon
out of the determinant
nn
co
7) 1iii RRR

 for all

,1,,inn2
8) The remaining determinant goes to “Dn” which is
equal to (−1)[n/2].
Hence proved for L-matrix.
The proof for rev-inv-L matrix is as follows:
5) 1iii
RRR

 for all

,1,,inn2
6) Take an, (a1 − a2), (a2 − a3), ... (an−1 − an) common
out of the determinant
7) 1iii
RRR

 for all

1, 2,,1in
8) The remaining determinant goes to “Dn” which is
equal to


2
1n 
Hence proved for rev-inv-L matrix also.
Therefore, from the above we can say that any L-fami-
ly matrix of order n*n will be a non-singular matrix if
and only if nth originator is non-zero and any ith generator
is not equal to to (i + 1)th originator (for all


1, 2,,1in

) matrix.
Claim 5: If we permute




ai with its adjacent
number i.e. with


1i
a or


1i
a (in circular way), the value of


DL
n changes to







11
1
nn
nn n
aaa
DL
aa a




 
and




 

11
12
n
n
aaa
DL
aaa

 
 
Proof 5:
Case 1. When replacing ai by ai‒1 for 2,3, ,in


and a1 by an (in Circular Manner)
i.e. 1nn aa

, and
122
,,
nn
aa a

1 a1n aa
then the value of Det. become




 

11
1
nn
nn n
aaa
D
aa a




 L
and by dividing it by the
actual value of


L,





11
1
nn
nnn
aaa
D
Daa a




Case 2. When replacing ai by ai + 1 for
and an by a1 (in Circular Manner)
1, 2,,1in
i.e. , and
12
aa23 1 ,,
n
aa aa

n 1n aa
then the value of Det. become




 

11
12
n
n
aaa
DL
aaa

 
 
and

R. GUPTA, G. R. MURTHY 20
by dividing it by the actual value of Det (L),




 
11
12
n
n
aaa
D
Daaa

 
Now note that, if we divide both the ratios,






11 2
11
n
nn
aaa
D
Daa a




 
3. Block “L” Matrix
If we take any one of the four kind of L matrix and make
a bigger matrix (having order greater than the previous
matrix) which contain the previous matrix then this type
of matrix can be characterized as Block “L” where the
Matrix has the same building block all over the matrix.
Let us consider a “L” matrix having minimum order (2
 2) –
ab
aa



Now, we are considering a shape where b = 0 then
0a
Laa





which has a shape of “└”
If we take this matrix and make a new matrix which
has this matrix as a building block then
0L
XLL



here L is the same as described above.
Here we can see that the value of


2
La
Again, if we can take X as a building block and if we
follow the same shape “└”, we can get a new matrix
0X
L
X
X




4
where “X” is as described above.
Note that, all the matrices are following the same pat-
tern and hence having a same shape.
If we calculate the Determinant of the above matrices
[5]:
 
22
X
LLaaa
Likewise, for Y,
 
44
YXXaa 
8
a
where the matrix Y
has order = 2  2  2 = 8
So we can generalized the det. value as


-Block n La
where n is the order of the matrix.
Now, if we more generalize our Block Matrices with
different L matrices having different elements, then we
can write
X
 as –
1
23
0L
LL





where 1 L

, 2 L

, 3 L

are L matrices having different ele-
ments.
Note,


2
11
La





2
22
La




2
33
La


and that is how, the value of
   
2
22
131313
XLLaaaa


Again, going for bigger ordered matrices, we have Y

which has blocks of
X

,
X
 ,
X
 and if we go
through above method, we can find the
 
22
13 13
YXXaabb

 

Where are elements of
123
,,,aaa
X
 matrix. like-
wise, are elements of
123
,,,bbb
X
 . Here we can
write
X

 as
1
23
0L
LL





 

So, In general, Determinant value of Block “L” matri-
ces can be written as:
 
222
13 13 13
Block Laabbcc
Where are elements of different “L” Ma-
trices.
,,, ,abcd
4. Hybrid “L” Matrix
We can make a matrix in which it has blocks of different
kinds of “L” matrices like └, ┘, ┐, ┌.
They may or may not repeat in the matrix. We are
calling this type of matrix as hybrid “L” matrix where the
building block of matrix is different types of L matrix.
This type of shapes can be found in the nature itself.
00
00
ab
aa bb
ccdd
cd






Matrix having all four type of L matrices.
Here we can see the different L patterns. The elements
are arranged in this fashion that they are constructing
different L shapes. In future, the Determinant value and
inverse of the above matrix can be evaluated.
5. Conclusion and Future Work
In this technical report, we reflect on the approach of
arriving at structured matrices. Specifically, we propose
some concrete approaches to define innovative structured
matrices. Furthermore, we define the family of L-matri-
ces and study some of their properties. We expect this
class of matrices to find many applications in future.
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R. GUPTA, G. R. MURTHY 21
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