LINEAR ALGEBRA OF PASCAL MATRICES

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1 LINEAR ALGEBRA OF PASCAL MATRICES LINDSAY YATES Abstract. The famous Pascal s triangle appears in many areas of mathematics, such as number theory, combinatorics and algebra. Pascal matrices are derived from this triangle of binomial coefficients, which create simplistic matrices with interesting properties. We explore properties of these matrices and the inverse of the Pascal matrix plus the identity matrix times any positive integer. We further consider a unique matrix called the Stirling matrix, which can be factorized in terms of the Pascal matrix.. Introduction The ancient arithmetic triangle, today known as Pascal s triangle, is an infinite numerical table represented in triangular form. The numbers displayed in the triangle are called binomial coefficients, ( n k), which represent the number of ways of picking k unordered outcomes from n possibilities. Each entry in the triangle is obtained by adding together two entries from the row above: the one directly to the left and the one directly to the right; this pattern can be seen in the image below. The Pascal s triangle has been known for over ten centuries. The set of numbers that form the Pascal s triangle were known before Blaise Date: December 5, 4.

2 LINEAR ALGEBRA OF PASCAL MATRICES Pascal, although he is attributed with being the first one to publish the information known about the triangle in his treatise, Traité du triangle arithmétique. The numbers originally arose from Indian studies of combinatorics and the Greeks interest in figurate numbers. These numbers were continually discussed by Islamic mathematicians during the th century and in the th century by a Persian poet named Omar Khayyam. They were also seen in China during the 3th century. The Pascal s triangle was officially published in Pascal s treatise soon after his death in 665. This triangle arises in many areas of mathematics such as algebra, probability, and combinatorics. We were motivated by the Pascal s triangle prominence in the field of mathematics and its many applications, in particular Pascal matrices. We wanted to further our studies to consider the various properties and unique connections that Pascal matrices has to other functions and number sequences.. Pascal Matrices The Pascal s triangle can be transcribed into a matrix containing the binomial coefficients as its elements. We can form three types of matrices: symmetric, lower triangular, and upper triangular, for any integer n >. The symmetric Pascal matrix of order n is defined by S n (s ij ), where ( ) i + j s ij for i, j,,..., n () j We can define the lower triangular Pascal matrix of order n by L n (l ij ), where l ij {( i ) if i j j otherwise The upper triangular Pascal matrix of order n is defined by U n (u ij ), where () u ij {( j ) if j i i otherwise (3)

3 LINEAR ALGEBRA OF PASCAL MATRICES 3 We notice that U n (L n ) T, for any positive integer n. For example, for n 5 we have: S L U These Pascal matrices have some interesting properties, which we present next. Theorem.. [] Let S n be the symmetric Pascal matrix of order n defined by (), L n be the lower triangular Pascal matrix of order n defined by (), and U n be the upper triangular Pascal matrix of order n defined by (3), then S n L n U n. Proof. Let L n be the lower triangular Pascal matrix of order n defined by () and U n be the upper triangular Pascal matrix of order n defined by (3). By direct multiplication of matrices L n and U n we obtain the ij-th element of the product L n U n : l ik u kj l ik l kj, since U n (L n ) T. k Then, k k l ik l jk k k ( )( ) i j k k j k j ( )( ) i j, since l k j k ik for k>j. The Vandermonde identity says that: t ( )( ) m n t n t ( m + n n ( )( ) i j k k ), for any m,n,t N (4) k Let m i, n j, and t k in (4). j ( )( ) ( ) i j i + j Then, s k j k j ij, the entries of the symmetric Pascal matrix S n. Hence, S n L n U n.

4 LINEAR ALGEBRA OF PASCAL MATRICES 4 This result can be used to determine the determinant of the symmetric Pascal matrix, S n. Corollary.. If S n is the symmetric Pascal matrix of order n defined by (), then det(s n ), for any positive integer n. Proof. Let S n be the symmetric Pascal matrix of order n defined by (). By Theorem., we know that S n L n U n, where L n is the lower triangular Pascal matrix of order n defined by () and U n is the upper triangular Pascal matrix of order n defined by (3). Since L n and U n are triangular matrices, then det(l n ) and det(u n ). It follows that det(s n ) det(l n U n ) det(l n )det(u n ). Definition.3. [5] Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that P AP B. Theorem.4. [] Let S n be the symmetric Pascal matrix of order n defined by (), then S n is similar to its inverse S n. This result shows the following property of the eigenvalues of S n. Corollary.5. [] Let S n be the symmetric Pascal matrix of order n defined by (). Then the eigenvalues of S n are pairs of reciprocal numbers. Proof. Let S n be the symmetric Pascal matrix of order n defined by () and λ be an eigenvalue of S n. Since the det(s n ), we know S n is invertible. It follows that λ and, λ is an eigenvalue of Sn. Since S n and Sn are similar by Theorem.4, then S n and Sn have the same eigenvalues. Hence, λ and λ are eigenvalues of S n, and the eigenvalues of S n are pairs of reciprocal numbers. Remark. If n is odd, since the eigenvalues must come in pairs, one of the eigenvalues must be equal to. Example.6. The eigenvalues of the symmetric Pascal matrix, S, are λ and λ 3 5, where λ λ gives a reciprocal pair. Example.7. For n odd, let n 3. Then the eigenvalues of the symmetric Pascal matrix, S 3, are λ 4+ 5, λ 4 5, and λ 3. We note that λ λ gives a reciprocal pair and λ 3 is a selfreciprocal.

5 LINEAR ALGEBRA OF PASCAL MATRICES 5 In their paper, A Note on Pascal s Matrix, Cheon, Kim, and Yoon found an interesting factorization of the lower triangular Pascal matrix, L n. ( ) In k O Theorem.8. [4] Let G k T O S k be a matrix of order n, where S k is the matrix of order k defined by: s ij { if i j j>i for every k,,..., n. Then the lower triangular Pascal matrix of order n can be written as: L n G n G n G. For example, L To further our studies of the lower triangular Pascal matrix, we are interested in studying the inverse of this matrix. Theorem.9. [6] Let L n be the lower triangular Pascal matrix of order n defined by (), then L n (( ) i j l ij ). Proof. We will show that L n L n I n. By direct multiplication of L n and L n we get the ij-th element of the product: ( ) k j l ik l kj. (5) k

6 LINEAR ALGEBRA OF PASCAL MATRICES 6 If i < j then the element (5) is zero and if i j, then the element (5) is. We will show that for i > j, the element (5) is zero. If i > j, the element is: ( )( ) ( )( ) i j i j ( ) j j j j i j( )( ) i i i j ( ) [ ( ) ( ) i i j i j j i j i j Hence, L n L n ( ) i j( i j ) ]. I n and L n (( ) i j l ij ) is the inverse of L n. There is another unique way in which the inverse of the lower triangular Pascal matrix can be written, using the Hadamard product of matrices. Definition.. [8] Let A, B be m n matrices. product of A and B is defined by: [A B] ij [A] ij [B] ij, for i m, j n. The Hadamard Theorem.. [8] Let τ n be a n n lower triangular matrix defined below as: { ( ) i j if i j, τ ij otherwise The inverse of the lower triangular matrix can be found using the Hadamard product: For example, if n 4, then: L n L n τ n L Inverse of the Pascal Matrix Plus An Integer In this section we are going to describe the inverse of L n + ki n where L n is the lower triangular matrix of order n defined by (), I n is the identity matrix and k is a positive integer. We call L n + ki n the Pascal matrix plus an integer. First, we are considering the case for k. By direct computation of the inverse of L n +I n, we can observe that there is

7 LINEAR ALGEBRA OF PASCAL MATRICES 7 a close relation between the inverse of L n +I n and the Pascal matrix L n. For example, for n 4, (L 4 + I 4 ) L 4 + I 4, In their paper, Explicit Inverse of the Pascal Matrix Plus One, S.L. Yang and Z.K. Liu showed that the inverse of L n + I n is the Hadamard product between L n and a lower triangular matrix. We are going to describe this unique lower triangular matrix next. For this we need to define the Euler polynomials. Euler polynomials, E n (x), can be defined by the following generating function: e tx e t + n The first few Euler polynomials are: E (x) E (x) x E (x) x x E 3 (x) x 3 3 x + 4 E n (x) tn n! Theorem 3.. [8] For n, the n n inverse matrix Q n (P n +I n ) is given by: q ij { ( i j ) Ei j () if i j, if i < j Proof. From the definition of Euler polynomials, E n, we have:

8 e tx e t + LINEAR ALGEBRA OF PASCAL MATRICES 8 E n (x) tn n!. n ( ) t n Then, (E n (x + ) + E n (x)) n! n ( ) ( ) t n E n (x + ) + t n E n (x) n! n! n et(x+) e t + + etx e t + ( ) t n e tx x n. n! n n Comparing the coefficients of tn n! in the above equation, we obtain: E n (x + ) + E n (x) x n, for n. (6) In [8] it was proved that for all n : From (6) and (8), we get: k Setting x in (7), we obtain: ( ) n E n (x + y) E k (x)y n k (7) k k ( ) n E n (x + ) E k (x) (8) k k ( ) n E k (x) + k E n(x) x n, for n. (9) In other terms: E n (y) k ( ) n E n k ()y k () k E n (x) k ( ) n E n k ()x k, for n () k

9 LINEAR ALGEBRA OF PASCAL MATRICES 9 Let E(x) and X(x) be column matrices defined by: E(x) [E (x), E (x),..., E n (x)] T and X(x) [, x,..., x n ] T. Let Ēn be the lower triangular n n matrix defined by: e ij From (9) and (), we get: {( i j ) Ei j () if i j, otherwise Then we obtain, (P n + I n )E(x) X(x), E(x) ĒnX(x). (P n + I n ) Ēn. A connection between Euler polynomials and Bernoulli numbers can be found. Bernoulli numbers, B k, can be defined by: t e t t n B n n n! Euler polynomials can be expressed in terms of Bernoulli numbers by the following expression: E n (x) n+ ( k+ ) ( ) n+ Bk x n+ k k n + k This connection between Euler polynomials and Bernoulli numbers gives another expression of the inverse of L n + I n. Theorem 3.. [8] For n, the n n inverse matrix Q n (P n +I n ) is given by: ( i )( i j+ )B i j+ if i j, q ij j i j + if i < j

10 LINEAR ALGEBRA OF PASCAL MATRICES We also note that the inverse of the Pascal matrix plus one can be written in terms of a Hadamard product. Theorem 3.3. [8] Let n be a matrix of order n defined as: or ij { E i j() if i j, if i<j ( i ) i j+ B i j+ if i j, ij j i j + if i<j Then (L n + I n ) L n n. We observe that the inverses of the matrices L n and L n +I n both can be written as a Hadamard product between L n and a lower triangular matrix. We now want to extend our scope of study to consider the inverse of L n +ki n, where k is a positive integer, to see if these inverses can also be expressed as a Hadamard product of L n and a lower triangular matrix. By direct computation of the inverse of (L n +ki n ), for k and k 3, we obtain for n 4: (L 4 + I 4 ) (L 4 + 3I 4 ) We note from these examples that the pattern is preserved. The inverse of L n + ki n can be written as L n n. The question now is what are the elements of the matrix n. In their paper, Inverting the Pascal Matrix Plus One, Aggarwala and Lamoureux found that these numbers are values of polylogarthmic functions

11 LINEAR ALGEBRA OF PASCAL MATRICES Definition 3.4. The polylogarithm function, denoted Li m (λ), is the function defined as: Li m (λ) k λ k k m λ + λ m + λ3 3 m +... where m is an integer and λ is a complex parameter. For integer values of the polylogarithm order, we have: Li (λ) ln( λ) Li (λ) λ λ λ Li (λ) ( λ) λ( + λ) Li (λ) ( λ) 3 The polylogarithmic functions satisfy the following recurrence relation: Li m λ Li m λ Using the result found by Aggarwala and Lamoureux in [], we can describe the inverse of the Pascal matrix plus an integer in terms of the Hadamard product. Theorem 3.5. Let L n be the lower triangular matrix of order n defined by (), and n be the n n lower triangular matrix defined by: ( ) i j+ k (Li j i( k)) if i>j ij if i j + k if i<j Then, (L n + ki n ) L n n, where k is a positive integer. 4. Stirling Matrix via Pascal Matrix Very similar to the Pascal matrix, the Stirling matrix can be defined using the Stirling numbers of the second kind. In this section we describe

12 LINEAR ALGEBRA OF PASCAL MATRICES a unique connection between Stirling matrices and Pascal matrices. Definition 4.. [4] The Stirling numbers of the second kind denoted S(n, k) for integers n and k, where k n, count the ways to divide a set of n objects into k nonempty subsets. Example 4.. Let X {a, b, c, d}, then the partitions for each k,, 3, 4: k : X; k : [{a}, {b, c, d}], [{b}, {a, c, d}], [{c}, {a, b, d}], [{d}, {a, b, c}], [{a, b}, {c, d, }], [{a, c}, {b, d}], [{a, d}, {b, c}]; k 3 : [{a}, {b}, {c, d}], [{a}, {c}, {b, d}], [{a}, {d}, {b, c}], [{c}, {d}, {a, b}], [{b}, {d}, {a, c}], [{b}, {c}, {a, d}]; k 4 : [{a}, {b}, {c}, {d}]. Thus, if k 7 if k S(4, k) 6 if k 3 if k 4 We can take these Stirling numbers of the second kind and transcribe them into a matrix, just as the binomial coefficients of the Pascal s triangle can be written as a Pascal matrix. Definition 4.3. Let S n be the Stirling matrix of order n, defined by: { S(i, j) if i j s ij otherwise For example, when n 4: S 4 3, 7 6 We note that the elements of the Stirling matrix satisfy the following recurrence relation: s ij s i,j + js i,j

13 LINEAR ALGEBRA OF PASCAL MATRICES 3 The Stirling matrix can be factorized via the Pascal matrix as the following: ( ) In k Theorem 4.4. [4] Let A k T P k be a matrix of order n, where P k is the lower triangular Pascal matrix of order k defined by (). Then the Stirling matrix of order n can be written as: S n A n A n A. Remark. When k n in Theorem 4.4, then A n is the lower triangular Pascal matrix. Example 4.5. The Stirling matrix of order 5 is: S Conclusion We see from our studies that the Pascal s triangle is very prominent in many areas of mathematics. The ability to transcribe the binomial coefficients into matrices that have many unique properties and connections is what has so many mathematicians interested and excited to study Pascal matrices. References [] A. Edelman and G. Strang, Pascal Matrices, The Mathematical Association of America, Monthly (4) [] R. Aggarwala and M. Lamoureux, Inverting the Pascal Matrix Plus One, The Mathematical Association of America, Monthly 9, No. 4 (April )

14 LINEAR ALGEBRA OF PASCAL MATRICES 4 [3] G. Call and D. Velleman, Pascal s Matrices, The Mathematical Association of America, Monthly (993), [4] G. Cheon, J. Kim and H.Yoon, A Note on Pascal s Matrix, Mathematics Subject Classification (999). [5] Lay, David C. (). Linear Algebra and Its Applications: 4th Ed. Boston: Pearson, Addison-Wesley. [6] Lawden, G.H. Pascal matrices, The Mathematical Gazette, [7] R. Brawer and M. Pirovino, The Linear Algebra of the Pascal Matrix, Elsevier Science Publlishing Co., Inc (99), 3-3. [8] S. Yang and Z. Liu, Explicit Inverse of the Pascal Matrix Plus One, International Journal of Mathematics and Mathematical Sciences (6) -7. Lindsay Yates, Mathematics Department, Georgia College, Milledgeville, GA 36, U.S.A. address:

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