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This is a talk on

The

Magnificent Matrix

and its next

Generation structures

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Delivered in the Spring Workshop on Combinatorics and Graph Theory,

2006

Held at

Center for CombinatoricsNankai University

TianjinPeoples’ Republic of China

on April 21, 2006

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By

Prof. R.N.MohanSir CRR Institute of Mathematics

Eluru-534007, AP, India

Andhra University

----------

Visiting ProfessorTWAS-UNESCO Associate member

Center for Combinatorics

Nankai University, Tianjin, PR China

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Magnificent Matrixotherwise 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

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The three types mentioned here are combinatorially equivalent

And each is useful in its own way for the

construction of many :

Combinatorial Configurations

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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.

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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)

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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

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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.

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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

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These designs are used

In

Communication & Networking systems

by

Charles Colbourn, Dinitz and Stinson.

Jointly and independently, and by many others also

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specifically Mn-matrices

• Gave the method of construction of

• μ-resolvable and

• Affine μ-resolvable BIB and

PBIB designs

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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 treatments in common

• If the blocks of different sets have treatments in common then they are called as affine resolvable designs

1q

2q

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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

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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

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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

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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

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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

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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,

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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.

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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

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The orthogonal numbers are defined by• The formula

1

( )n

l m i ii

g R R rs

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A property

1

2

, 1,

,

3, 2 , 0,1,2,...,

2

j

i i

i n i

R R

R R n

nR R n where i

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Sum

• The sum of the orthogonal numbers

is given by

(n+1)/2

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Because it is given by

• By the formula

1/ 2

0

4 2 ( 1) / 2n

k

k n n

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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.

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Why they miss????

We consider the sum of all orthogonal

numbers including these

missing numbers

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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

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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.

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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

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Example

• From Mn = [aij],

• where aij = 1 + (i-1)(j-1) mod n,

• n is a prime

• i, j = 1, 2, 3,4,5.

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Mn-matrix

• Is given by11111

12345

13524 .

14253

15432

nM

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M-matrix of Type I

• Is given by

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

M

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SPBIB design

• Is given by

• v = 4 = b, r = k = 2, λ1 = 1, λ2 = 0.

• The solution is

1 3 1 2

3 4 2 4

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M-Graph (next generation)

• And the graph is

Elements

Columns5 6 7 8

1 2 3 4

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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

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For n = 11

• We get an M-graph as

Elements

Columns

101 2 3 4 5 6 7 8 9

101 2 3 4 5 6 7 8 9

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M-matrix Type II

• When n + 1 is a prime,

• Take aij = (i j) mod (n+1), i, j = 1,2,...,n

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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.

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The sum of orthogonal numbers

• Is given by

2/ 22

1 0

' are the orthogonal numbers

then their sum is (4 ) 0

i

nn

ii k

If g s

g k n

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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.

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Graph

• The existence of an M-matrix of type II,

• implies the existence of

• a Regular Bipartite Graph.

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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

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M-Graph

• Its regular bipartite graph is as follows:

Elements

Columns1 2 3 4 5 6

1 2 3 4 65

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These M-graphs give

A new family of fault-tolerant M-networks

• We will show some of its features

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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

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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

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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.

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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

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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).

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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.

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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.

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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.

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Trivial orthogonal number

•

,i iR R n

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The sum of the orthogonal numbers

• Is given by

1

2

1

n

ii

g

1

2

1

4 2 ( 1) / 2

n

k

k n n

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Determinant

• | M | = 1

121 2n

n

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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.

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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)

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Example

• Take n = 9.

• Then from the equation (i+j) mod n

• we get the Mn-matrix as

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Mn-matrix

• That is

21

3

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67

98

41 2 3 5 6 7 8 9

2 3 4 5 6 7 8 9 13 4 5 6 7 8 9 1 24 5 6 7 8 9 1 2 35 6 7 8 9 1 2 3 46 7 8 9 1 2 3 4 57 8 9 1 2 3 4 5 68 9 1 2 3 4 5 6 79 1 2 3 4 5 6 7 81 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.

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Useful

• But these types of matrices are very useful in digital image processing.

• Reference:

• Mathematica: Digital Image Processing

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M-matrix of Type III

• It is given by

21

3

54

67

98

4

-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 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 -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

1 2 3 5 6 7 8 9

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Incidence matrix / adjacency matrix

• It is1 2 3 4 5 6 7 8 9

9

678

12345

0 1 0 1 0 1 0 1 11 0 1 0 1 0 1 1 00 1 0 1 0 1 1 0 11 0 1 0 1 1 0 1 00 1 0 1 1 0 1 0 11 0 1 1 0 1 0 1 00 1 1 0 1 0 1 0 11 1 0 1 0 1 0 1 0

1 0 1 0 1 0 1 0 1

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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.

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M-graph

• Is given by

Elements

Columns

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

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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

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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

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Hence

• it is called as

Magnificent matrix

Mn- matrix

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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.

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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

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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 non-orthogonal.

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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.

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Non-orthogonal

• The non-orthogonal property is

1, , 1,2,..., .

2 2i j k

n nR R g for k or

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question• Is there any matrix such that,

, , i jR R c

where c is a constant for all its rows.

If so what is the method of construction

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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.

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Just for curiosity sake

• I conclude by quoting a magic square

Sum of the entries

• in any direction turns out to be 15

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Magic square

4 9 2

3 5 7

8 1 6

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Lo Shu

• This is first appeared in Chinese literature in

• third millennium BC

• In cabbalistic and occult literature.

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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.

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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.

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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).

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continued

9.Jafarkhani, H.(2001) A quasi orthogonal space-time 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).

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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.

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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

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Questions if any, please

Thank You all