Permutation Generation Method on Evaluating Determinant of Matrices
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1 Article International Journal of Modern Mathematical Sciences, 2013, 7(1): International Journal of Modern Mathematical Sciences Journal homepage: ISSN: X Florida, USA Permutation Generation Method on Evaluating Determinant of Matrices Haslinda Ibrahim *, Lugean M. Zake, Zurni Omar School of Quantitative Sciences, College of Arts and Sciences, Universiti Utara Malaysia, Sintok Kedah, Malaysia *Author to whom correspondence should be addressed; Tel.: ; Fax: Article history: Received 26 February 2013, Received in revised form 5 June 2013, Accepted 6 June 2013, Published 7 June Abstract: This paper presents a new method for finding determinant for any size of matrices. The fundamental feature of this algorithm is that it uses the idea of permutation generation. A novel combinatorics approach of generating permutation based on starter sets is demonstrated. This permutation algorithm will be a basis to develop a method to calculate determinant. We provide a numerical example for a matrix of size 4 x 4 to illustrate the algorithm. Eventually, general framework for evaluating determinant via this combinatorics approach will be presented. Keywords: Permutation, determinant, starter set. Mathematics Subject Classification (2010): 05A05, 15B35, 15B36 1. Introduction The study of determinants has a long history and appears to be essential in some geometric and algebraic computations. There are various alternative methods to compute determinant of matrices [2,3,7,9,12,13]. These methods can be classified into two categories namely direct method and nondirect methods. In a direct method, a specific mathematical formula is used to obtain determinant of matrices without converting the original matrix into other matrix forms. Examples of direct method are
2 13 cross multiplication, cofactor expansion, condensation method and permutation. Whilst for non-direct methods the matrix is factorized into some different forms. An example of non-direct method is the Gauss elimination [10]. Furthermore another method to find the determinant of matrices is using combinatorial approach, namely permutation algorithms. The algorithm of permutation obtains the determinant of matrices from its definition by generating all the permutations and copying the entries according to each permutation. Each permutation can also be used to provide a sign of the product. Furthermore, the execution time and the complexity of the entire algorithm for determinant of matrices can be reduced by employing permutation. Hence, permutation method can be considered the most favourable algorithms among other algorithms for determinants of matrices [7,8]. Based on the advantages of permutation method over the other methods in determining determinant, this combinatorial approach will be employed in calculating determinant. Thus, this paper will provide the appropriate permutation method to be the basis in evaluating determinant. General framework for this new algorithm will be developed. A numerical example of a 4 x 4 matrix will be used to illustrate the algorithm. 2. Combinatorial Approach in Generating Permutation Permutation is different arrangements that can be made with a given number of things by taking some or all of them at a time. In general, considerable work has already been done on algorithms listing distinct n! permutations [1,6,12]. In this section we provide algorithm for generating permutation based on starter set [5]. This algorithm will be the catalyst to develop new algorithm in finding determinant Generating Permutation by Using Starter This section provides algorithm for generating all permutations by using combinatorial approach [5]. Notably, this new algorithm for listing all permutations for n elements was developed based on distinct starter sets. Hence, once the starter sets are obtained, each starter set is then cycled to obtain the first half of the distinct permutations. The complete list of permutations is achieved by reversing the order of the first half of the permutation. This algorithm has advantages over the other algorithms due to its simplicity and ease of use [5]. Therefore this algorithm is employed in this study to construct new algorithm for determinant of matrices. The followings definitions and theorems are necessary for developing a new algorithm for evaluating determinant [5]. Definition 2.1. A starter set is a set that is used as a basis to enumerate other permutations.
3 14 Definition 2.2. A reversed (inverse) set is a set that is produced by reversing the order of permutation set. Definition 2.3. An equivalence starter set is a set that can produce the same permutation from any other starter set. Theorem 2.4. There exist starter sets for each n elements for listing all n! permutations. Theorem 2.5. There exist n cycles for each starter sets of n elements. Theorem 2.6. There exists n! permutations from starter sets. We provide a numerical example for listing 4! distinct permutations. Here are the following steps: Step 1: Fix one element from the integers (1, 2, 3,4), for example 1, we have the following starter sets permutation: (1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4), (1, 3, 4, 2), (1, 4, 2, 3), (1, 4, 3, 2) Step2: Find all equivalent starters sets permutations and starter sets in the following: (1, 2, 3, 4) (1, 4, 3, 2) (1, 2, 4, 3) (1, 3, 4, 2) (1, 3, 2, 4) (1, 4, 2, 3) Step 3: Delete the equivalence starter sets permutation, we produce the following: (1, 2, 3, 4) (1, 2, 4, 3) (1, 3, 2, 4) Step 4: Generate all permutations from these starters set {(1, 2, 3, 4), (1, 2, 4, 3), (1, 3, 2, 4)}. All permutations are generated cyclically for every permutation as the following: (1, 2, 3, 4), (2, 3, 4, 1), (3, 4, 1, 2), (4, 1, 2, 3), (1, 2, 4, 3), (2, 4, 3, 1), (4, 3, 1, 2), (3, 1, 2, 4), (1, 3, 2, 4), (3, 2, 4, 1), (2, 4, 1, 3), (4, 1, 3, 2). Step 5: Find the inverses (reverse) of the above permutations,
4 15 (4, 3, 2, 1), (1, 4, 3, 2), (2, 1, 4, 3), (3, 2, 1, 4), (3, 4, 2, 1), (1, 3, 4, 2), (2, 1, 3, 4), (4, 2, 1, 3), (4, 2, 3, 1), (1, 4, 2, 3), (3, 1, 4, 2), (2, 3, 1, 4). The general algorithm can be found in [5]. This algorithm will be a basis to generate starter sets and its application in calculating the determinant for n n matrices. 3. A New Algorithm for Evaluating Determinant This section presents results for the algorithm development. Then a new algorithm for finding determinant will be developed. In addition, a proof for the new algorithm for determinant of matrices will be presented. Definition 3.1. Let be the inverse permutation, then the sign of, denoted by sign is defined by sign( 1 ) ( 1) 1-1 number of inversion pair sin 1-1 if the number of inversion in is even if the number of inversion in -1 is odd Moreover, we call an even permutation if the sign = 1, and we call an odd permutation if the sign = 1. Definition 3.2. Let be the permutation, Then the sub-determinantis a calculation by summation diagonal with adding sign and inverse diagonal with adding sign for every sub matrices of permutations. Definition 3.3. Let be the permutation, then the equivalent permutation is inverse of permutation after fixing the first element of permutation. Before explaining the new algorithm for determinant, we present the following proposition, which will be used in developing the new algorithm for determinant. Proposition3.4. Let P be a cycle permutation, where P = ( n 1n), and U= (1 n n ), then P U. Proof. P = ( n 1 n) (1) ( n 1 n) P is cycle permutation
5 16 (2 3) (3 4) (4 5) (5 6) (n 1 n) (n 1 n) (5 6) (4 5) (3 4) (2 3), (n n 1) (6 5) (5 4) (4 3) (3 2). (n n ) = (1) (n n ) = (1 n n ) U. Now, we provide the numerical example to calculate determinant using permutation based on starter set. The numerical example is given to illustrate steps of this new algorithm for a matrix of size 4 4. Given a matrix A The determinant of A with the new algorithm can be obtained by generating all permutations based on starter set. Recall we have three starter set: (1, 2, 3, 4), (1, 3, 4, 2) and (1, 4, 2, 3). Then use each starter set and permutation for these starters to evaluate determinant. Figure 1 provides sub matrices from first starter set (1, 2, 3, 4). Then, use these permutations to generate sub matrices from original matrix by exchanging columns of matrix as the followings: Figure 1. Calculating sub-determinants from (1, 2, 3, 4) starter set From this starter set we produce the following result: (-72+18) + (-21+75) + (125-28) + (24 120) = 1
6 17 Figure 2. Calculating sub-determinants from (1, 3, 4, 2) starter set From starter set (1, 3, 4, 2) we produce the following result: (15-105) + ( ) + ( ) + (84 15) = -51 Figure 3. Calculating sub-determinants from (1, 4, 2, 3) starter set Finally from starter set (1, 4, 2,3) we have the following result: ( ) + (36-36) + (42 30) + ( ) = 32 Now we gather all the sub determinants of the sub matrices with each other and we add sign of permutations and sign of inverses permutation to find the determinant for original matrix, as show below:
7 18 det(a) = (( 72+18) ( 21+75) + (125 28) (24 120)) + ((15 105) ( )+ ( ) (84 15)) + (( 60+60) (36 36) + (42 30) ( 50+70)) = 18. Now, we are ready to provide the general framework of this new algorithm to evaluate determinant. Suppose for any n n matrices, we can find determinant by using the following steps. Step 1: Determine starter sets for n elements. Step 2: Use every column of permutations from step 1 to generate all sub matrices [ ] [ ] [ ] [ ] Step 3: Find the sub determinant for the sub matrices which are produced in step 2 by using the following: a. Multiply diagonal elements and multiply inverse diagonal elements for every sub matrix produced from step 4, such as... [ ] [ a 1n. a n1 a 11 a 22 a 33 a nn a 11 a n2 a 12 a 23 a 34 a n1 a 1(n-1) a nn a 1n a 21 a 32 a n(n-1) ]
8 19 [ ] [ a 1n a n1 a 11 a 23 a 32 a nn a 11 a n3 a 13 a 22 a 34 a n1 a 1(n-1) a nn a 1n a 21 a 33 a n(n-1) ] for all sub matrices. b. Find the sign for permutation, and add to the diagonal elements. So the sign permutation is used with multiplication of diagonal elements, and the sign for inverse permutation is used with multiplication of inverse diagonal elements as the following: c. Calculate the sub determinants of the sub matrices by sum diagonal and inverse daigonal for every submatrices. Step 4: Combine all the products of the sub determinants for all sub matrices which are produced from step 3. Step 5: The final result for the determinant to the original matrix is det(a) = sign a 11 a 22 a 33 a nn + signa 1n a n1 + sign a 12 a 23 a 34 a n1 +sign a 11 a n2 + + signa 1n a 21 a 32 a n(n-1) +sign a 1(n-1) a nn From this algorithm we will be able to prove the following a new formula for determinant. Theorem We have ( )
9 20 Proof. Case 1:Suppose A is a matrix of size 2 2, and there are two permutations 1 (1, 2), 2 (2, 1). We take only permutation 1(1, 2) and delete permutation 2(2, 1) because this permutation is the inverse of 1(1, 2). Then, take the matrix based on the permutation, such as A. We can use the following expression to create the intersection of the diagonals of the one matrix (1) then we get from matrix A,a 11 a 22 +a 12 a 21. By using the definition of sign, and inversion and by adding the sign to the permutations resulting from the previous expression, we get a 11 a 22 + ( 1) a 12 a 21 = a 11 a 22 a 12 a 21. So, (1) becomes Then, we use this equation to find the determinant of matrix for one permutation only. This equation for one permutation without inverse takes one matrix. So that a solution for the determinant of matrix A for the permutation (1,2) is given by det (A) = a 11 a 22 a 12 a 21. Case 2: Consider a matrix A of size 3 3, we get 3! = 6 permutations of {(1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), (3, 2, 1)},which are generated from (1, 2, 3) without repetition and inverse. The permutations here have only 3 permutations which are {(1, 2, 3), (2, 3, 1), (3, 1, 2)}. Take the matrices for these permutations after changing the columns for the original matrix as follows: For the permutation (1, 2, 3) take a matrix A 1, for the permutation (2, 3, 1) take a matrix A 2, and the permutation (3, 1, 2) take a matrix A 3. To find the determinant for every sub matrix from matrix A, use equation (2) above and multiply the diagonal elements and multiply the inverse diagonal elements and take the sign a diagonal and inverse diagonal, such as:
10 21 Combining these equations we get Then = + + [ ] [ ] ( ) Case 3: Now, we expand this operation on the matrix size n n, so we get ( ) permutation without repetition and we get ( ) sub matrices based on the permutations. Therefore, the algorithm for the determinant for a matrix A which is
11 22 Determine by using the equation (2) for every sub matrices produced from A after swapping of permutation by exchanging columns so that a sub matrix A 1 comes from a permutation ( n 3 n 2 n 1 n) such as A 1 = Then we make swap in a permutation to get a sub matrix A 2, A 2 = Repeat this step to all permutations to get all sub matrices whose number is ( ) from A based on swapping in permutations. Now, by using equation (2), for every sub matrix, the determinant for these matrices is as follows: To find the determinant of the matrix A, we just need to combine every determinant of sub matrix, to obtain =
12 23 Therefore, we get (( ) ( ) ) This equation is produced by combining determinants for all sub matrices. We have ( ) sub matrices produced from ( ) permutations without repeating matrix size n n. Then ( )
13 24 4. Conclusion We have proposed new method to evaluate determinant based on starter set from listing permutation. We shall consider the following questions for future study: Could we construct isomorphism classes to classify equivalence starter sets? We expect to use one-factorization or decomposition of compete graph (K n ). Could we compare this new algorithm for finding determinant with other existing methods? Could we employ these new algorithms (permutation based on starter set and diagonal method for determinant)in other fields, for example in cryptography(security), image processing (determinant), and medicine (permutation)? Acknowledgments We are indebted to Ministry of Higher Education for providing this grant (Code #:11768). References [1] C. T. Fike, A permutation generation method. The Computer Journal, 18(1) (1975): [2] E. H. Bareiss, Sylvester s identity and multistep integer-preserving Gaussian elimination. Mathematics of Computation, 22(103) (1968): [3] F. Benno, Modified Gauss algorithm for matrices with symbolic entries. ACM Communications in Computer Algebra, 42(3) (2008): [4] G. I. Mani, Permutation generation using matrices. Retrieved on November 01, 2010, from [5] H. Ibrahim, Z. Omar, & A. M. Rohni, New algorithm for listing all permutations. Modern Applied Science, 4(2)(2010): [6] J. S. Rohl, Generating permutations by Choosing. The Computer Journal, 21(4) (1978): [7] K. Thongchiew. A computerized algorithm for generating permutation and its application in determining a determinant. World Academy of Science, Engineering and Technology, 27(2007): [8] M. Mahajan &V. Vinay, Determinant: Combinatorics, algorithms, and complexity. Chicago Journal of Theoretical Computer Science,5(1997):
14 25 [9] M. L. Griss, An efficient sparse minor expansion algorithm. In J. Gosden, & O. G. Johnson (Ed.), ACM National Conference 76, Proceedings of the association for computing machinery annual conference, ACM, New York, NY, 1976, p [10] O. Rezaifar & M. Rezaee, A new approach for finding the determinant of matrices. Applied Mathematics and Computation, 188(2007): [11] R. Sedgewick, Permutation generation methods. Journal Computer Science of Applied Mathematics, 9(1977): [12] T. Sasaki & H. Murao, Efficient Gaussian elimination method for symbolic determinants and linear systems. ACM Transactions on Mathematical Software (TOMS), 8(3)(1982): [13] W. M. Gentleman & S.C. Johnson, The evaluation of determinants by expansion by minors and the general problem of substitution. Mathematics of Computation, 28(126)(1974):
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