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 coeffi- cients, which create simplistic matrices with interesting properties. We explore properties of these matrices and the inverse of the Pas- cal 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. 1. Introduction The ancient arithmetic triangle, today known as Pascal's triangle, is an infinite numerical table represented in triangular form. The num- n bers displayed in the triangle are called binomial coefficients, 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, 2014. 1 LINEAR ALGEBRA OF PASCAL MATRICES 2 Pascal, although he is attributed with being the first one to publish the information known about the triangle in his treatise, Trait´edu tri- angle arithm´etique. 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 10th century and in the 11th century by a Persian poet named Omar Khayyam. They were also seen in China during the 13th cen- tury. The Pascal's triangle was officially published in Pascal's treatise soon after his death in 1665. 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 applica- tions, 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. 2. 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 > 0. The symmetric Pascal matrix of order n is defined by Sn = (sij), where i + j − 2 s = for i; j = 1; 2; ::::; n (1) ij j − 1 We can define the lower triangular Pascal matrix of order n by Ln = (lij), where (i−1 j−1 if i ≥ j lij = (2) 0 otherwise The upper triangular Pascal matrix of order n is defined by Un = (uij), where ( j−1 if j ≥ i u = i−1 (3) ij 0 otherwise LINEAR ALGEBRA OF PASCAL MATRICES 3 T We notice that Un = (Ln) , for any positive integer n. For example, for n = 5 we have: 0 1 1 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1 1 1 B 1 2 3 4 5 C B 1 1 0 0 0 C B 0 1 2 3 4 C B C B C B C S5 = B 1 3 6 10 15 C L5 = B 1 2 1 0 0 C U5 = B 0 0 1 3 6 C B C B C B C @ 1 4 10 20 35 A @ 1 3 3 1 0 A @ 0 0 0 1 4 A 1 5 15 35 70 1 4 6 4 1 0 0 0 0 1 These Pascal matrices have some interesting properties, which we present next. Theorem 2.1. [1] Let Sn be the symmetric Pascal matrix of order n defined by (1), Ln be the lower triangular Pascal matrix of order n de- fined by (2), and Un be the upper triangular Pascal matrix of order n defined by (3), then Sn = LnUn. Proof. Let Ln be the lower triangular Pascal matrix of order n defined by (2) and Un be the upper triangular Pascal matrix of order n defined by (3). By direct multiplication of matrices Ln and Un we obtain the ij-th element of the product LnUn: n n X X T likukj = liklkj; since Un = (Ln) . k=1 k=1 n n j X Xi − 1j − 1 Xi − 1j − 1 Then, l l = = = ik jk k − 1 k − 1 k − 1 k − 1 k=1 k=1 k=1 j Xi − 1j − 1 = , since l = 0 for k>j: k − 1 j − k ik k=1 The Vandermonde identity says that: n X m n m + n = ; for any m,n,t 2 (4) t n − t n N t=0 Let m = i − 1, n = j − 1, and t = k − 1 in (4). j Xi − 1j − 1 i + j − 2 Then, = = s , the entries of the sym- k − 1 j − k j − 1 ij k=1 metric Pascal matrix Sn. Hence, Sn = LnUn. LINEAR ALGEBRA OF PASCAL MATRICES 4 This result can be used to determine the determinant of the symmetric Pascal matrix, Sn. Corollary 2.2. If Sn is the symmetric Pascal matrix of order n defined by (1), then det(Sn) = 1, for any positive integer n. Proof. Let Sn be the symmetric Pascal matrix of order n defined by (1). By Theorem 2.1, we know that Sn = LnUn, where Ln is the lower triangular Pascal matrix of order n defined by (2) and Un is the upper triangular Pascal matrix of order n defined by (3). Since Ln and Un are triangular matrices, then det(Ln) = 1 and det(Un) = 1. It follows that det(Sn) = det(LnUn) = det(Ln)det(Un) = 1. Definition 2.3. [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 −1AP = B. Theorem 2.4. [1] Let Sn be the symmetric Pascal matrix of order n −1 defined by (1), then Sn is similar to its inverse Sn . This result shows the following property of the eigenvalues of Sn. Corollary 2.5. [1] Let Sn be the symmetric Pascal matrix of order n defined by (1). Then the eigenvalues of Sn are pairs of reciprocal numbers. Proof. Let Sn be the symmetric Pascal matrix of order n defined by (1) and λ be an eigenvalue of Sn. Since the det(Sn) = 1, we know Sn −1 −1 is invertible. It follows that λ 6= 0 and, λ is an eigenvalue of Sn . −1 −1 Since Sn and Sn are similar by Theorem 2.4, then Sn and Sn have −1 the same eigenvalues. Hence, λ and λ are eigenvalues of Sn, and the eigenvalues of Sn are pairs of reciprocal numbers. Remark 1. If n is odd, since the eigenvalues must come in pairs, one of the eigenvalues must be equal to 1. Example 2.6.p The eigenvaluesp of the symmetric Pascal matrix, S2, 3 + 5 3 − 5 are λ = and λ = , where λ λ = 1 gives a reciprocal 1 2 2 2 1 2 pair. Example 2.7. For n odd, let n = 3. Thenp the eigenvaluesp of the sym- metric Pascal matrix, S3, are λ1 = 4+ 15; λ2 = 4− 15; and λ3 = 1. We note that λ1λ2 = 1 gives a reciprocal pair and λ3 = 1 is a self- reciprocal. 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, Ln. T In−k O Theorem 2.8. [4] Let Gk = OSk be a matrix of order n, where Sk is the matrix of order k defined by: ( 1 if i ≥ j s = ij 0 j>i for every k = 1; 2; :::; n. Then the lower triangular Pascal matrix of order n can be written as: Ln = GnGn−1 ··· G1. For example, 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 B 1 1 0 0 0 C B 0 1 0 0 0 C B 0 1 0 0 0 C B 0 1 0 0 0 C B C B C B C B C L4 = B 1 1 1 0 0 C B 0 1 1 0 0 C B 0 0 1 0 0 C B 0 0 1 0 0 C B C B C B C B C @ 1 1 1 1 0 A @ 0 1 1 1 0 A @ 0 0 1 1 0 A @ 0 0 0 1 0 A 1 1 1 1 1 0 1 1 1 1 0 0 1 1 1 0 0 0 1 1 01 0 0 0 01 B1 1 0 0 0C B C = B1 2 1 0 0C B C @1 3 3 1 0A 1 4 6 4 1 To further our studies of the lower triangular Pascal matrix, we are interested in studying the inverse of this matrix. Theorem 2.9. [6] Let Ln be the lower triangular Pascal matrix of or- der n defined by (2), then −1 i−j Ln = ((−1) lij). −1 Proof. We will show that Ln · Ln = In. −1 By direct multiplication of Ln and Ln we get the ij-th element of the product: n X k−j (−1) liklkj: (5) k=1 LINEAR ALGEBRA OF PASCAL MATRICES 6 If i < j then the element (5) is zero and if i = j, then the element (5) is 1.