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This is a talk on
The
Magnificent Matrix
and its next
Generation structures
1
Delivered in the
Spring Workshop on Combinatorics and Graph Theory,
2006
Held at
Center for Combinatorics
Nankai University
Tianjin
Peoples’ Republic of China
on April 21, 2006
2
By
Prof. R.N.Mohan
Sir CRR Institute of Mathematics
Eluru-534007, AP, India
Andhra University
---------Visiting Professor
TWAS-UNESCO Associate member
Center for Combinatorics
Nankai University, Tianjin, PR China
3
Magnificent Matrix
otherwise called as
• Mn-Matrix is a square matrix obtained by:
(aij) = (di x dh dj) mod n,
by suitably defining di , dh , dj , x in different ways.
For example:
• 1. 1+(i-1)(j-1) mod n (for n is a prime)
• 2. (i.j) mod n (for n or n+1,is a prime)
• 3 (i+j) mod n (for n is a positive integer)
• Still there are so many ways to explore
4
The three types mentioned here
are combinatorially equivalent
And each is useful in its own way for the
construction of many :
Combinatorial Configurations
5
The combinatorial configurations
mainly are
• Balanced Incomplete Block (BIB) Designs
• Partially Balanced Incomplete Block
(PBIB) Designs
• Symmetric BIB and PBIB designs
• Graphs
•
Latin squares, orthogonal arrays, sub arrangements, Youden squares etc.
6
The Mn-Matrices
•
•
•
•
•
•
•
Gives rise to Mn-Graphs, defined as
If given an Mn-matrix:
Ck’s be its columns
aij’s be its elements
let V1 = {Ck}, V2 ={aij} be the vertex-sets
An edge is αijk iff aij is in Ck.
This gives the Mn-graph (V1, V2, αijk)
7
LDPC code
• By using the pattern of Mn-matrix
aij = 1+(i-1)(j-1) mod n
• Bane Vasic and Ivan of Arizona, USA
Constructed
Low-density Parity Check (LDPC) Codes
8
These Mn-matrices
• Have been used in the construction of
these BIB and PBIB designs
• A BIB designs, is an arrangement in which
• v elements are arranged in b blocks,
• each element is coming in r blocks
• and each block is having k elements
• and each pair of elements is coming in λ
blocks.
9
If λ is not constant
• Then they are called as:
Partially balanced incomplete block designs
• If v = b and r = k then the design is called
• Symmetric design
10
These designs are used
In
Communication & Networking systems
by
Charles Colbourn, Dinitz and Stinson.
Jointly and independently, and by many others also
11
specifically Mn-matrices
• Gave the method of construction of
• μ-resolvable and
• Affine μ-resolvable BIB and
PBIB designs
12
Affine Resolvability,
Resolvability
• If the b blocks are grouped in to t sets of m
blocks each then the design is said to be
Resolvable
• If the blocks of the same set have q
1
treatments in common
• If the blocks of different sets have q
2
treatments in common then they are called as
affine resolvable designs
13
Application
• Thus when blocks are grouped into parallel
classes then the resolvability exist in a design,
limited block intersection leads to affine nature.
• These classes are called resolution classes
• If the set of m messages assigned to a
particular user forms a parallel class or
resolution class
14
Then comes the next generation
• These Mn-matrices lead to the construction of
Three types of M-matrices (The next Generation)
• namely:
• Type I is with 1+(i-1)(j-1) mod n,Prime
• Type II is with (i.j) mod n (n+1 prime)
• Type III is with
(i+j) mod n (n is an integer)
• And their corresponding M-graphs
15
Those are defined as
• M-matrix of Type I
• Definition. When n is a prime,
• consider the matrix of order n obtained by
the equation
• Mn = (aij), where
• aij = 1 + (i-1)(j-1) mod n, when n is prime
where i, j = 1, 2,.., n
16
M-matrix of Type I
•
•
•
•
•
•
•
In the resulting matrix
retain 1 as it is
substitute -1’s for odd numbers
substitute +1’s for even numbers.
This gives M-matrix of Type I.
This is a symmetric n x n matrix.
Roles of +1 and -1 can be inter-changed
17
Hadamard matrix
• A matrix H having
• All +1’s in the first row and first column
• HH′= nIn
• It is an orthogonal matrix
• This is an important matrix having many applications
18
Resemblances and Differences
between M-Matrix & Hadamard Matrix.
• Both have +1’s in the first row and first column
• Both consist of +1 or -1 only
• Row sum in (M) is 1 and in (H) is zero
• (M) Exists for all primes, (H) exists for n =2 or 0 mod 4
• Both useful for the constructions of codes, graphs, and
designs,
and Sequences and array sequences
• (M) is Non-orthogonal, (H) is orthogonal,
19
Properties of M-matrix of Type I
• in each row and column,
the number of +1’s is (n+1)/2
• and the number of -1’s is (n-1)/2.
20
The orthogonal numbers are
• the orthogonal number between any two rows
• is given by
4k+2-n,
• where k is the number of +1’s in the selected set
21
The
orthogonal numbers are defined by
• The formula
n
g  ( Rl Rm )   ri si
i 1
22
A property
R1 , R j  1,
Ri , Ri  n
R2i , Rni
n3
 2  n, where i  0,1, 2,...,
2
23
Sum
• The sum of the orthogonal numbers
is given by
(n+1)/2
24
Because it is given by
• By the formula
n 1/ 2
 4k  2  n  (n  1) / 2
k 0
25
Here is an open problem
• Do all orthogonal numbers as per the
formula exist in a matrix concerned now?
• For example when n = 11, the orthogonal
numbers are -9, -5, -1, 3, 7, 11.
• But -5 and 7 do not exist. They are called
missing orthogonal numbers.
26
Why they miss????
We consider the sum of all orthogonal
numbers including these
missing numbers
27
Determinant
• Given an M-matrix of Type I
|M| = - 4 if n = 3
=
0 if n ≥ 5,
In an (1,-1)-matrix, when the determinant is
maximum then it is called as
Hadamard Matrix
28
SPBIB design
• The existence of an M-matrix of order n,
where n is a prime, implies the
• existence of an SPBIB design with
parameters
• v = b = n-1, r = k = (n-1)/2,
• λi vary from 0 to (n-3)/2.
29
Graph
• The existence of an M-matrix of Type I
• implies the existence of
• A regular bipartite graph
V = 2n (V1= n, V2=n), E = 2n
valence is (n-1)/2
30
Example
• From Mn = [aij],
• where aij = 1 + (i-1)(j-1) mod n,
• n is a prime
• i, j = 1, 2, 3,4,5.
31
Mn-matrix
• Is given by
11111 
12345 


M n  13524  .


14253 
15432 
32
M-matrix of Type I
• Is given by
1 1 1 1 1
1 1  1 1  1


M  1  1  1 1 1 .


1 1 1  1  1


1

1
1

1
1


33
SPBIB design
• Is given by
• v = 4 = b, r = k = 2, λ1 = 1, λ2 = 0.
• The solution is
1 3
3 4
1
2
2
4
34
M-Graph (next generation)
• And the graph is
5
6
7
8
1
2
3
4
Columns
Elements
35
Usable
• These types of graphs form a new family
of graphs, which are highly usable in
• routing problems of
• salesmen, transportation or
• Communication and network systems
36
For n = 11
• We get an M-graph as
1
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
8
9
10
Columns
Elements
37
M-matrix Type II
• When n + 1 is a prime,
• Take aij = (i j) mod (n+1), i, j = 1,2,...,n
38
Orthogonal numbers
• orthogonal number between any two rows
Is given by
•
g = 4k-n
where k is the number of 1’s in the selected set.
39
The sum of orthogonal numbers
• Is given by
If
gi ' s are the orthogonal numbers
n2
2
then their sum is
n/2
 g   (4k  n)  0
i 1
i
k 0
40
SPBIB design
• The existence of an M-matrix of type II,
implies the existence of an SPBIB design
with parameters
• v = n= b, r =k = n/2,
• λi values vary from 0 to (n-3)/2.
41
Graph
• The existence of an M-matrix of type II,
• implies the existence of
• a Regular Bipartite Graph.
42
For n+1 = 7
• The SPBIB design, is given by
•
•
•
•
1 4 1 2 1 2
3 5 4 3 2 4
5 6 5 6 3 6
where as v = b = 6, r = k = 3, λ1= 2, λ2 = 1,
λ3 = 0, n1 = 2, n2 = 2, n3 = 1
43
M-Graph
• Its regular bipartite graph is as follows:
1
2
3
4
5
6
3
4
5
6
Columns
Elements
1
2
44
These M-graphs give
A new family of
fault-tolerant M-networks
• We will show some of its features
45
The main features of M-networks
• The maximum diameter of the M-network
is found to be 4 independent of the
network size.
• M-networks out-perform other known
regular networks in terms of throughput
and delay.
• exhibit higher degree of fault-tolerance
• as these graphs have good connectivity
46
Reliability
• they provide a reliable communication
system
• These networks are found to be denser
than many known multiprocessor
architectures
• such as mesh, star, ring, the hypercube
47
Lastly another application
There are n nodes in the network, and they
are to be inter-connected by using Buses.
A Bus is a communication device, which
connects two or more nodes and provides a
direct connection between any pair of nodes
on the bus.
48
M-matrix of Type III
• This matrix is obtained by (i+j) mod n
• When n is an integer odd or even
• Not necessarily prime
49
In similar way
In the resulting matrix
substitute
1 for even numbers and -1 for odd numbers and
also for 1,
( or 1 for odd numbers keeping the 1 in the matrix
as 1 itself and -1 for even numbers).
50
M-matrix of Type III
Then this resulting matrix M is called as
M-matrix of Type III.
When n is odd,
in each row and each column:
the number of +1’s is (n+1)/2
the number of -1’s is (n-1)/2.
51
When n is even
in each row (column) consists of
equal number of +1’s and -1’s
numbering to n/2 .
This is also a non-singular symmetric nxn
binary matrix.
52
When n is odd
• the orthogonal number
between any two rows
• is given by
4k-2-n,
• where k is the number of unities
• in the selected set.
53
Trivial orthogonal number
•
Ri , Ri  n
54
The sum of the orthogonal
numbers
• Is given by
n 1
2
 gi 
i 1
n 1
2
 4k  2  n  (n  1) / 2
k 1
55
Determinant
• |M|=
 1
n 1
2
2
n 1
56
When n is even
• In the case of even number
• the formula for the orthogonal number is
the same as in Type II as
• 4k-n
• and all the other treatment will follow.
57
Further study omitted
• And hence further study is not needed
• even though the structures of the matrices
are different as
• one generated from (i j)mod n,(n+1) is prime
• and the other generated from (i+j)mod n, (even)
58
Example
• Take n = 9.
• Then from the equation (i+j) mod n
• we get the Mn-matrix as
59
Mn-matrix
• That is
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
8
9
1
2
3
4
5
6
7
9
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
9
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circulant matrices
The circulant matrix is defined as n x n matrix
whose rows are composed of cyclically shifted
versions of a length n and a list ℓ.
And the list ℓ may consist of any elements like
(a1, a2 ,…,an)
on which no property was defined
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governed by
• But the matrix defined here is governed by
• (i+j) mod n,
• where n is an integer odd or even and
i, j = 1,2,…, n.
62
Useful
• But these types of matrices are very useful
in digital image processing.
• Reference:
• Mathematica: Digital Image Processing
63
M-matrix of Type III
• It is given by
1
2
3
4
5
6
7
8
9
1
-1
1
-1
1
-1
1
-1
1
1
2
1
-1
1
-1
1
-1
1
1
-1
3
-1
1
-1
1
-1
1
1
-1
1
4
1
-1
1
-1
1
1
-1
1
-1
5
-1
1
-1
1
1
-1
1
-1
1
6
1
-1
1
1
-1
1
-1
1
-1
7
8
9
-1
1
1
-1
1
-1
1
-1
1
1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
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Incidence matrix / adjacency matrix
• It is
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
0
1
0
1
0
1
0
1
1
1
0
1
0
1
0
1
1
0
0
1
0
1
0
1
1
0
1
1
0
1
0
1
1
0
1
0
0
1
0
1
1
0
1
0
1
1
0
1
1
0
1
0
1
0
0
1
1
0
1
0
1
0
1
1
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
65
SPBIB design
•
•
•
•
•
•
•
•
•
2 1 2 1 2 1 2 1 1
4 3 4 3 4 3 3 2 3
6 5 6 5 5 4 5 4 5
8 7 7 6 7 6 7 6 7
9 8 9 8 9 8 9 8 9
parameters
v = b = 9, r = k = 5, λ1=1, λ2 = 2, λ3 = 3, λ4 = 4.
n1=2, n2 = 2, n3 = 2, n = 2,
which is a 4-associate class SPBIB design.
66
M-graph
• Is given by
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
Columns
Elements
67
M-network system
• This type of M-graphs are found to be
highly fault tolerant,
• richly connected architectures
• for being a Network system
• usable in
• multi-processor and
• communication systems
68
Thus
• These types of M-matrices
have remarkable properties and features
• Available for the construction of
• Combinatorial designs &
• For applications in
Communication and Network Systems
69
Hence
• it is called as
Magnificent matrix
Mn- matrix
70
Originated from Mn-matrix
• The matrices and graphs
• Originated from Mn-matrices
• The three types of M-matrices &
• The three types of graphs
• Are its next generations.
71
The classes of orthogonalities
•
Different types of orthogonalilities:
1.
Orthogonal if the inner product of any distinct
rows of the matrix is 0
2.
Quasi Orthogonal matrix in which the columns are
divided into groups. The columns within each
group are not orthogonal to each other but
different groups are orthogonal to each other.
This has been used in coding theory by Zafarkhani
72
Semi-orthogonal
• If A is a real m × n matrix, where m ≠ n, such
that AA′ = Im or A′A= In, but not necessarily
both, is called semi-orthogonal matrix.
• n x 2n matrix, in which n x n matrix is
orthogonal and another n x n matrix is nonorthogonal.
73
Non Orthogonal
• Non-orthogonality is not of much
important as it seems because of
accountability of inner products.
• In our M-matrices that has been
accounted for.
74
Non-orthogonal
• The non-orthogonal property is
n 1 n
Ri , R j  g k , for k  1, 2,...,
or .
2
2
75
question
• Is there any matrix such that,
Ri , R j
 c,
where c is a constant for all its rows.
If so what is the method of construction
76
Without confusion
• Sylvester matrix has a constant row sum,
• Our M-matrices have row sums as 1or 0
•
(Except the first row in Type I)
• But we want a constant orthogonal
numbers all g’s should be constant.
• In Hadamard matrix all g’s are 0.
77
Just for curiosity sake
• I conclude by quoting a magic square
Sum of the entries
• in any direction turns out to be 15
78
Magic square
4
3


8

9 2

5 7
1 6

79
Lo Shu
• This is first appeared in Chinese literature in
• third millennium BC
• In cabbalistic and occult literature.
80
References
1. Colbourn, Charles (2000). Applications of combinatorial
designs in communications and networking, MSRI, Project
2000.
2. Colbourn, C.J., Dinitz, J.H., and Stinson, D.R. Applications of
Combinatorial Designs to Communications, Cryptography, and
Networking (1999). Surveys in Combinatorics, 1993, Walker
(Ed.), London Mathematical Society Lecture Note Series 187,
Cambridge University Press.
3. Fan, P., and Darnell, M. (1996). Sequence design for
communications applications. Research studies Press Ltd.
John Wiley & sons Inc.
81
continued
4. Ehlich, H. (1964). Determinantenabschätzungen für
binäre Matrizen." Math. Z. 83, 123-132
5. Ehlich, H. and Zeller, K. (1962).
Binäre Matrizen." Z. angew. Math. Mechanik 42, T20-21.
82
continued
6. Mathematica: Digital Image Processing
7. Teague, M. R. (1979)
Image analysis via the general theory of moments.
J. Optical Soc. America, 70(8):pp. 920-930,
8. Mohan, R .N., and Kulkarni, P.T. (2006).
A new family of fault-tolerant M-networks,
(IEEE, Trans. Computers, revision submitted).
83
continued
9. Jafarkhani, H.(2001) A quasi orthogonal spacetime block code. IEEE,Trans. Commu. 49, 1-4.
10. Chang, Yangbo Hua, Xiang-Gen Xia and Brian
Sudler. (2005). An insight into space-time block
codes using Hurwitz-Randon families of matrices
(Personal communication, (IEEE, Trans.
Information Theory, submitted).
84
This work can be seen at the
websites
11. Mohan, R.N., Sanpei Kageyama, Moon
Ho Lee, and Gao Yang. (2006). Certain
new M-matrices and applications.
Submitted to Linear Algebra and
Applications on April 10,06 and reference
no is LAA # 0604-250B, at this paper can
be viewed at
http://arxiv.org/abs/cs.DM/0604035, as an
e-print.
85
And
12. Mohan, R.N., Moon Ho Lee and Ram
Paudal. (2006) An M-matrix of Type III and
its Applications. Submitted to Linear
Algebra and Applications, on April 11,2006
with refe.No. LAA # 0604-253B and can be
viewed at
http://arxiv.org/abs/cs.DM/0604044, as an
e-print
86
Questions if any, please
Thank You all
87