The

(1)

Contents lists available at www.innovativejournal.in

ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS

Journal homepage: http://www.innovativejournal.in/index.php/ajcem

– MATRIX COMPLETION PROBLEM FOR DIGRAPHS OF

MATRICES

WITH LESS THAN FOUR DIRECTED LINES

Njoroge James Mwangi1_{,Kivunge Benard Kivunge}2_{, Muondwe Samuel Kanyi}3_,Kinyanjui_{Anthony muthondu}4

1, 4_{School of Mathematics, Technical university of Kenya, Nairobi, Kenya}

2, 3 _{Kenyatta University, Department of mathematics, Nairobi, Kenya}

ARTICLE INFO ABSTRACT

Corresponding Author Samuel Kanyi Muondwe Jomo Kenyatta University of agriculture and technology Nairobi Kenya

Key words: – matrix completion, digraphs of , less than four directed lines, non negative

A real matrix is said to be –matrix if its principal minors are non-negative and all its non-diagonal entries are non-positive. In this project, we consider the –matrix completion problem. It is shown that any digraph with less than four directed lines which is not a cycle has –completion. It is further shown that digraph of matrix with less than four directed lines which is a cycle does not have –completion.

AMS Subject Classification: 0070

A matrix is an ordered set of numbers listed in a rectangular array. A matrix has elements (entries) which may be numbers or any abstract quantities that can be added, multiplied and decomposed in various ways. This makes a matrix a key concept in linear algebra and matrix theory.

If a matrix has rows and columns then we say it is a square matrix. A diagonal matrix is a square matrix with all the non-diagonal elements zero. When all the elements of a matrix are zero, we call a zero matrix. Let a pattern for matrices is a subset of . A partial matrix is a matrix in which some entries are specified while others are free to be chosen. A partial matrix specifies a pattern if its specified entries lie exactly in those positions listed in the pattern. Let be a class of matrices. A pattern is said to have completion if every partial matrix specifying the pattern can be completed to a -matrix. A matrix completion problem ask whether a given partial matrix has a completion of a desired type e.g. the positive definite completion problem ask which partial Hermilitian matrices have a positive definite completion. Matrix completion problems arise in application whenever a full set of data is not available, but it is known that the full matrix data must have certain properties. Let be an matrix and an integer with a minor of is the determinant of matrix obtained from by deleting

rows and columns. A completion is said to be zero completion if all the unspecified entries in the partial matrix are substituted with zero. Matrix completion has a variety of applications such as statistics i.e. entropy methods for missing data. It can also be applied in Chemistry that is molecular conformation problems as well as within matrix theory for instance determinant inequalities. Completion problems have provided an

excellent mechanism for understanding matrix structure more deeply.

R. Grone, C. R. Johnson and others (1984) studied the positive completion of partial Hermitian matrices. V. I. Paulsen, S. C. Power and others (1989) studied the Schur products and matrix completions. In the same year, a minimum Rank completion for Block matrices was studied by H. J. Woerdeman. C. R. Johnson (1990) provided a survey of matrix completion problems focusing on the positive definite completion problem, rank completion and contraction completion. I. Gohberg, L. Rodman (1992) studied the Bounds for eigenvalues and singular values of matrix completions. M. Ba Kanyi and C. R. Johnson (1995) studied the Euclidean Distance Matrix completion problems. C. R. Johnson and R. L. Smith (1996) studied the completion problem for matrices and inverse M-matrices.

In recent years, graphs and digraphs have been successfully used in matrix completion problems. In this case patterns that are positionally symmetric have been studied by means of their graphs for example, inverse M-matrices and P-M-matrices. For patterns without positional symmetry, direct graphs (digraphs) have been used.

L. Hogben (2001) studied “The Graph Theoretic Methods for Matrix completion problems”. L. Hogben, and L. Dealba (2002) studied “The P-matrix Completion”. L. Hogben, B. Kivunge and others (2003) worked on “The non-negative P0-matrix Completion problem” using graph theoretic method. A Wangness (2005) studied matrix completion problems regarding various classes P0-matrices. J. Mutembei (2007) studied “M0-matrix completion problems for digraphs of and

matrices”. Mutembei established that all matrices of digraphs of matrices have M0-completion. She also had shown that all partial matrices of digraphs of

(2)

M0-completion. Also digraph of a which is either a cycle or clique does not have M0-completion. R. Kabusia (2008) studied “The M0-matrix completion problem for digraphs of

matrices for and 3”. P. Waweru (2008) studied “The M0-matrix completion problem for (selected

matrices) digraphs. Waweru established that for selected matrices, cases which are cycles have no zero completion while those which are not cycles have zero completion.

Our main objective is to classify all digraphs with 5 vertices, therefore we use strategies that are necessary for completing a partial M0-matrix. Thus we illustrate the strategies with partial M0-matrices. For M0-matrices, we require that all determinants to be non-negative. Definition 1.2.0

A square matrix is called symmetric if it is equal to its transpose i.e. for all and .

Definition 1.2.1

A submatrix of of matrix , denoted

, is obtained from by deleting all rows and columns of other than rows

and columns (Howard, 2005). If is matrix, the _{cofactor of} _denoted

where is the matrix obtained from by deleting row and column . If is an

matrix, a subset of with elements and is a subset of with elements, then we write for the minor of that corresponds to the rows with index in and the columns with index in If , then

is called a principal minor, that is a determinant of a principal submatrix. If the matrix that corresponds to a principal minor is a quadratic upper-left part of the larger matrix, then the principal minor is called a leading principal minor for subset of , a principal submatrix is obtained from by deleting all rows and columns not in

and is denoted by . A principal subpattern is obtained from a pattern by deleting all positions whose first or second co-ordinate is not in

A real matrix is called if all of its principal minors are non-negative. A partial

is a partial matrix in which all fully specified principal submatrices are . A non-negative

whose entries are non-negative. These entries must be specified, Choi, (2002).

A real matrix is called if all its principal minors are negative and all of its non-diagonal entries are non-positive, that is for all

. Therefore is a whose non-diagonal entries are non-positive. A partial

is a partial matrix in which all fully specified

principal submatrices are (Hogben, 2003).

Completion of a partial matrix is called zero completion if all the unspecified entries in the partial matrix are substituted with zeros. A pattern is said to be positively symmetric if it has the property that is the pattern and only if is also in the pattern.

A graph is a finite non-empty set of positive integers whose members are called vertices and a set of (unordered) pairs of vertices called edges of . If is an edge of , then we say

that and are adjacent in and (v, u) is incident with both and . The order of is the number of vertices of . A subgraph of a graph is a graph

where is a subset of and

is a subset of

A digraph is a finite set of positive integers , whose members are called vertices, and a set of ordered pairs of vertices called arcs (also called directed edges). In the arc is the tail and is the head.

A subdigraph of the digraph in a digraph where is a subset of

and is a subset of whose endpoint are in . A subdigraph is complete if it contains all possible arcs between its vertices (Hogben, 2001).

A path in a digraph (respectively graph) is a sequence of arcs in which vertices are distinct (except that possibly the first vertex is the same as the last). The length of a path is the number of arcs (edges) in the path. A cycle is a path with the last vertex equal to the first. A digraph is strongly connected if there is a path from any vertex to any other vertex. A graph is connected if there is a path from any vertex to any other vertex. A (sub) digraph is called a clique if it contains all possible edges between its vertices (Brualdi, 1991).

Two digraphs are said to be isomorphic if one can be obtained from the other by relabelling the vertices, that is, if there is a one to one correspondence between the vertices of the two digraphs. Two or more arcs joining the same pair of vertices are called multiple edges and an edge joining a vertex to itself is called a loop.

LEMMA 2.1

A pattern that includes all its diagonal position has P0-completion. A real matrix is P0-matrix if every principal minor is non-negative. We use this lemma since we know that a P0-matrix has the property that every principal minor is non-negative; which is a necessary condition for M0-completion.

CASE 2.2.1

Consider the digraph represented below.

Let

be a partial

M0-matrix representing the digraph above.

By Lemma 2.1, has P0-completion. By definition of completion, from definition 1.2.2, has P0-completion therefore,

Consider the following submatrix and their determinants. Submatrix Determinant

1.

. .

(3)

We solve for which are not specified by substituting them with zero, that is setting all the unspecified entries to zero,

, we have

1. det = .

. .

. . . .

25. det = 26. det = The determinants are negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph. Case 2.2.2

Consider the diagraph represented below.

Let

be a partial

By lemma 2.1 has P0-completion. By definition of completion, from definition 1.2.2 has P0-completion therefore,

The determinants are the same as those in case 2.2.1 except where appears. We only consider these cases.

Setting the unspecified entries to zero, Submatrix Determinant 1. 11. . . . .

We solve , which are not specified by substituting them with zero, that is setting all the unspecified entries to zero,

, we have

1. det =

. . .

26. det = =

The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph.

Case 2.2.3

Let

be a partial

By Lemma 2.1, has P0-completion. By definition of completion, from definition 1.2.2 has P0-completion therefore,

The determinants are the same as those in case 2.2.1 except where and appears. We only consider these cases.

Submatrix Determinant 1. 4.

. . .

, we have

1. det =

. .

26. det =

=

The determinants are non-negative and non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph.

Case 2.2.4

Let

be a partial

(4)

Submatrix Determinant 1.

. .

.

, we have

1. det = since the submatrix is fully

specified

. .

26. det =

since the

submatrix

is fully specified

The submatrix is fully specified because and are already specified.

Case 2.2.5

Let

be a partial

. .

, we have

4. det = 5. det =

. .

25. det = 26. det = The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the diagraph.

Case 2.2.6

Let

be a partial

The determinants are the same as those in case 2.2.1 except where , and appears. We only consider these cases.

. .

, we have

1. det =

5. det =

. .

25. det = 26. det = The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph.

Case 2.2.7

Let

be a partial

(5)

The determinants the same as those in case 2.2.1 except where , and appears. We only consider these cases.

.

. . . . .

, we have

27. det = 8. det =

. .

26. det =

Case 2.2.8

Let

be a partial

6. 8.

. .

, we have

6. det =

8. det =

. .

26. det =

Case 2.2.9

Let

be a partial

The determinants the same as those in case 2.2.1 except where , and appears. We only consider these cases. We only consider these cases.

. .

, we have

1. det = 4. det =

. .

26. det =

The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph

Case 2.2.10

Let

be a partial

(6)

. .

, we have

1. det = (since

the submatrix is fully

specified).

. .

26. det =

Since the

submatrix

is fully specified

The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph

Case 2.2.11

Let

be a partial

Submatrix Determinant 1.

. .

, we have

1. det = (since

the submatrix is fully

specified).

. .

25. det =

26. det =

since the submatrix

is fully specified The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph.

Case 2.2.12

Consider the diagraph represented below. 1



5 2

 

4 3

Let

be a partial

.

. . .

, we have

1. det =

4. det =

. .

20. det = 21. det = The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.4, there is zero completion; hence from definition 1.2.3 there is M0-matrix completion for the digraph.

Case 2.2.13

Consider the diagraph represented below. 1



(7)

 

4 3

Let

be a partial

and , (not zero)

We show that does not have M0-completion by a counter example as shown below.

det =

=

Since .

This implies that is not an M0-matrix. Hence there is no M0-completion for the digraph.

Conclusion

All digraphs for matrices with less than four directed lines which are neither cycles nor cliques have M0-matrix completion. However all digraphs for matrices of order

with less than four directed lines which are cycles or cliques have no M0-completion. Therefore there is no zero completion in this case.

In this paper, we have considered non-isomorphic digraphs for matrices of order with less than four directed lines. We have established that all the partial matrices of digraphs with less than four directed lines which are not cycles have an M0-completion. We have also shown that any

partial matrix which is a cycle does not have M0-completion.

ACKNOWLEDGEMENTS

I am deeply indebted to my devoted supervisor Dr. B. Kivunge for the assistance accorded through his tireless guidance through this project. He was a great source of inspiration and a great teacher. I appreciate S. Muondwe and A. muthondu for their contributions. My colleagues; Siro, Laban, Kemboi and Gauki for their company and endless support. Special thanks to my wife and my daughters, Anne and Mary for their support and continuous encouragement to work hard. My appreciation to my other family members for their support and all the others not mentioned who contributed in any way to my life when doing this work. All glory, honour and praise to the Most High God my Saviour.

REFERENCES

1. Brualdil, R. A. and Ryser, H. J., Combinatory Matrix Theory, Cambridge University Press, Cambridge, 1991. 2. Chartand, G., Oellermann, O. R., Applied and Algorithmic

Graph Theory.

3. Choi, J. Y., Dealba, L. M., Hogben L., Maxwell M. S. and Wangness, A., The P0-Matrix Completion Problem,

Electronic Journal of Linear Algebra, Vol. 9; 1-20, 2002. 4. Hogben, L., Graph Theoretic Methods for Matrix

Completion Problems, Linear Algebra and its Applications, 328; 161-202, 2001.

5. Hogben, L., Kivunge, B. et al, The Non-Negative P0

-Matrix Completion Problem, Electronic Journal of Linear Algebra Vol. 10, 46-59, 2003.

6. Johnson, C. R., Matrix Completion Problem, a Survey Proceedings of Sumposia of Applied Mathematics 40, American Mathematical Society, Providence, RI, 171-198, 1990.

7. Kabusia, R. M., The M0-Matrix Completion Problem for

Digraphs of Matrices for , Kenyatta University, Kenya, 2008.

8. Munakata, J., Matrices and Linear Programming. Holden-day Inc Sanfrancisco, 1979.

9. Mutembei, J., M0-Matrix Completion Problem for

digraphs and Matrices. Kenyatta University, Kenya, 2007.

10. Van Lint, J. H. and Wilson, R. M., A Course in Combinatorics, Cambridge University Press, 2001. 11.

12. Waweru, P., The M0-Matrix Completion Problem for (selected matrices) digraph; Jomo Kenyatta

University of Agriculture and Technology, Kenya, 2008.

CASE 2.2.1 Submatrix

Determinant

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

(8)

20.

21.

22.

23.

24.

25.

26.

We solve for which are not specified by substituting them with zero, that is setting all the unspecified entries to zero, , we have

2. det =

3. det =

4. det =

5. det =

6. det =

7. det =

8. det =

9. det =

10. det =

11. det =

12. det =

13. det =

14. det =

15. det =

16. det =

17. det =

18. det =

19. det =

20. det =

21. det =

22. det =

23. det =

24. det =

25. det =

26. det =

27. det =

CASE 2.2.2

Submatrix Determinant

1.

11.

12.

13.

21.

(9)

22.

23.

26.

We solve , which are not specified by substituting them with zero, that is setting all the unspecified entries to zero, , we have

2. det =

11. det =

12. det =

13. det =

21. det =

22. det =

23. det =

26. det =

=

CASE 2.2.3

1.

4.

11.

12.

13.

15.

16.

21.

22.

23.

26.

(10)

2. det =

4. det =

11. det =

12. det =

13. det =

15. det =

16. det =

21. det =

22. det =

23. det =

26. det =

=

CASE 2.2.4

1.

11.

12.

13.

21.

22.

23.

26.

2. det = since the submatrix is fully

specified 11. det =

since the submatrix is fully specified

12. det = since the submatrix

is fully specified

21. det =

since the submatrix

is fully specified

22. det =

since the submatrix

is fully specified

is fully specified 26. det =

since the submatrix

is fully specified CASE 2.2.5

(11)

4.

5.

11.

13.

15.

16.

17.

18.

21.

22.

23.

24.

25.

26.

6. det =

7. det =

11. det =

13. det =

15. det =

16. det =

17. det =

18. det =

21. det =

22. det =

23. det =

24. det =

25. det =

26. det =

The determinants are non-negative and the non-diagonal entries are non-positive. From definition 1.2.5, there is zero completion, hence from definition 1.2.4 there is M0-matrix completion for the diagraph.

CASE 2.2.6

1.

5.

8.

11.

12.

13.

14.

(12)

21.

22.

23.

24.

25.

26.

2. det =

5. det =

8. det =

12. det =

13. det =

14. det =

17. det =

18. det =

20. det =

21. det =

22. det =

23. det =

24. det =

25. det =

26. det =

CASE 2.2.7

6.

8.

10.

12.

14.

16.

17.

19.

20.

21.

23.

24.

(13)

25.

26.

8. det =

9. det =

10. det =

12. det =

14. det =

16. det =

17. det =

19. det =

20. det =

21. det =

23. det =

24. det =

25. det =

26. det =

CASE 2.2.8

6.

8.

10.

12.

14.

16.

17.

19.

20.

21.

23.

24.

25.

26.

(14)

6. det =

8. det =

10. det =

12. det =

14. det =

16. det =

17. det =

19. det =

20. det =

21. det =

23. det =

24. det =

25. det =

26. det =

CASE 2.2.9

1.

4.

5.

8.

11.

12.

13.

14.

15.

16.

17.

20.

21.

22.

23.

24.

25.

26.

2. det =

5. det =

8. det =

11. det =

(15)

13. det =

14. det =

15. det =

16. det =

17. det =

20. det =

21. det =

22. det =

23. det =

24. det =

25. det =

26. det =

CASE 2.2.10

1.

8.

11.

12.

13.

14.

17.

20.

21.

22.

23.

24.

25.

26.

1. det = (since the submatrix is fully

specified). 8. det =

11. det =

(since the submatrix is fully specified).

12. det =

= (since the submatrix

is fully specified). 13. det = =

is fully specified). 14. det =

17. det =

20. det =

21. det =

(16)

22. det =

is fully specified).

23. det =

25. det =

26. det =

since the submatrix

1.

8.

11.

13.

17.

21.

22.

23.

25.

26.

1. det = (since the submatrix is fully

specified). 8. det =

11. det =

(since the submatrix is fully specified).

12. det =

is fully specified). 13. det = =

18. det =

20. det =

21. det =

22. det =

23. det =

(17)

since the submatrix

1.

4.

7.

11.

12.

13.

16.

19.

21.

22.

23.

24.

25.

26.

1. det =

5. det =

6. det =

11. det =

12. det =

13. det =

15. det =

16. det =

19. det =

21. det =

22. det =

23. det =

24. det =

25. det =

26. det =

CASE 2.2.13

We show that does not have M0-completion by a counter example as shown below.

det =

=