Available online at wwwsciencedirectcom Linear Algebra and its Applications 429 (2008 50 56 wwwelseviercom/locate/laa Matrices with convolutions of binomial functions and Krawtchouk matrices orman C Severo State University of ew York at Buffalo USA Received 20 December 2006; accepted 5 February 2008 Available online 7 March 2008 Submitted by RA Brualdi Abstract We introduce a class S of matrices whose elements are terms of convolutions of binomial functions of complex numbers A multiplication theorem is proved for elements of S The multiplication theorem establishes a homomorphism of the group of 2 by 2 nonsingular matrices with complex elements into a group G contained in S As a direct consequence of representation theory we also present related spectral representations for special members of G We show that a subset of G constitutes the system of Krawtchouk matrices which extends published results for the symmetric case 2008 Elsevier Inc All rights reserved AMS classification: 5A8; 5A24; 5A99; 20C99; 33C47 Keywords: Matrices; Convolution of binomial functions; Multiplication theorem; Representation theory; Krawtchouk matrices The matrix systems S and G For integers M 0 and for complex numbers α and β let A(α β; M denote the + bym + matrix with (i j element {( M α a ij (α β; M i j M i+j β i j M i j 0 0 elsewhere Address: 50 Meadowview Lane Williamsville Y 422 USA Tel/fax: + 76 634 4248 E-mail address: severo@buffaloedu 0024-3795/$ - see front matter ( 2008 Elsevier Inc All rights reserved doi:006/jlaa200802002
C Severo / Linear Algebra and its Applications 429 (2008 50 56 5 ( for i 0 and j 0M Corresponding to each 2 by 2 matrix with complex elements we define {( } which we denote by asthe + by + matrix whose { jth} column j 0 is the matrix product A(α β; ( j A(γ ; j0for example equals for 0 the ordinary 2 by 2 matrix for and for 2 α 2 α 0 ( 0 0 γ 2 α 2 αγ γ 2 2αβ ( β α γ 0 0 2γ 2αβ βγ + α 2γ β 2 0 β 0 0 2 β 2 β 2 Let S denote the set of + by + matrices where α β γ and are indexed over all complex numbers We define G to be that subset of S such that each element of G satisfies α βγ / 0 We shall prove a multiplication theorem valid for all elements of S and apply this theorem to establish a homomorphism between the group of 2 by 2 nonsingular matrices with complex elements and G Using results of representation theory we also present a simple spectral representation for special members of G Furthermore we shall show that subsets of G constitute systems of Krawtchouk matrices which extends the intensive treatments in [345] for the symmetric case 2 The multiplication theorem { } For i j 0 the (i j element of is j λ ij (α β γ ; a iv (α β; ja ν0 (γ ; j0 ν0 U(ij vl(ij ( ( j j α i j+v β i v γ j v v (2 i v v where L(x y max(0x+ y and U(xy min(x y Ifα β and γ are all nonzero then we may write the sum on ν simply from 0 to either i or j We note that for any complex k and k 2 λ ij (k α k βk 2 γk 2 ; k j k j 2 λ ij (α β γ ; (22 and λ ij (k α k 2 βk γk 2 ; k i k2 i λ ij (α β γ ; (23 We can also easily show that for each fixed j 0 λ ij (α β γ ; (α + β j (γ + j (24 i0 By applying (23 to(24 we get the generating function of λ j for fixed j 0 viz λ ij (α β γ ; u i (α + βu j (γ + u j (25 i0
52 C Severo / Linear Algebra and its Applications 429 (2008 50 56 which exhibits the convolution nature of the λ j functions In passing we note that an alternative definition of λ ij (α β γ ; is simply the coefficient of u i in the expression (α + βu j (γ + u j However this definition did not lend itself to proving Theorem below Lemma For M 0 and complex numbers α β γ a and b A(a b; M A(αa + γbβa+ d; M Proof The (i j element on the left side of Eq (26is τ ij λ iμ (α β γ ; a μj (a b; M μ0 M+j μ0 U(iμ νl(iμ M (26 ( ( ( μ μ M α i μ+ν β i ν γ μ ν ν a M μ+j b μ j i ν ν μ The corresponding element on the right side of Eq (26 is i U(μj ( M τ ij (αa + γb M i+μ (βa + d i μ i μ μm+i νl(μj ( M j μ ν ( i ν Using the binomial expansion we get α M μ j+ν β μ ν γ j ν ν (27 τ ij S Kα i j h+ν β i k ν γ j+h ν k+ν a M h k b h+k where K ( ( ( M M j i i μ μ ν ν ( M i + μ h ( i μ k and the sum is over the set S of points (μνhk satisfying: μ M + i i; ν L(μjU(μj; h 0 M i + μ; and k 0i μ If we equate the coefficients of α c β c 2γ c 3 c 4a c 5b c 6 then that term of τ ij in which μ μ and ν ν corresponds to those terms of τ ij satisfying μ ν h ν + j ν k + ν and μ h + k + j (For example the first of these three equations arises both from equating the exponent c of τ ij and τ ij and the exponent c 3 of τ ij and τ ij For fixed μ and ν let S(μ ν denote that subset of S such that μ h + k + j and ν k + ν We note that S(μ ν S(μ ν if either μ μ or ν ν and S(μ ν S where the union is over integer values of μ and ν between 0 and Thus our problem reduces to proving ( ( ( μ μ M i ν ν μ K (28 j S(μ ν
C Severo / Linear Algebra and its Applications 429 (2008 50 56 53 The right side of Eq (28 reduces to M j μ0 ν0 ( ( ( M M j j i μ μ ν ν ( M i + μ μ ν j + ν which by using the identity ( ( ( ( ( ( r r r 2 r2 r r r 4 r3 r 2 r 3 r 4 r 4 r 3 r 2 r 4 r 4 ( i μ ν ν with r M r 2 i μ r 3 μ i and r 4 ν ν is easily shown to be equal to ( M j ( j μ μ j ( j M ( M μ ( + j M j ν ν ν i ν + ν μ μ ν ν0 μ0 Thus by applying the Vandermonde convolution formula we obtain the left side of Eq (28 This completes the proof of the lemma We are now prepared to prove our principal result concerning the multiplication of two elements of S Theorem The set S is closed under matrix multiplication and furthermore {( ( } a c a c αa + γb αc+ γd b d b d βa + b βc + d Proof For j 0 column j of the left side of Eq (29 is A(a b; ja(c d; j0 which by successively applying the lemma yields A(αa + γbβa+ d; j A(c d; j0 j A(αa + γbβa+ b; ja(αc + γdβc+ d; j0 (29 The latter expression is by definition the jth column of the right-hand side of (29 This completes the proof The next two results follow directly from Theorem Corollary G is a non-abelian group under matrix multiplication with unit 0 0 Corollary 2 Eq (29 expresses a homomorphism of the group of 2 by 2 nonsingular matrices into G A direct consequence of representation theory provides a spectral representation for special members of G
54 C Severo / Linear Algebra and its Applications 429 (2008 50 56 Theorem 2 If f (λ (α λ( λ βγ has distinct zeros λ and λ 2 then β { λ 0 {V } {V 0 λ 2 } } (20 where V is the 2 by 2 matrix whose jth column is an eigenvector of eigenvalue λ j j 2 ( corresponding to the Proof From the conditions given we may write ( ( λ 0 V V 0 λ 2 Applying Theorem and noting that {V } G we obtain { {V } λ 0 {V } diag(λ 0 λ 2 } j λ j 2 j 0 (2 Therefore λ j λ j 2 j 0 are the eigenvalues of in some order and the jth column of {V } is an eigenvector corresponding to λ j λ j 2 Finally we have {V } {V } which follows directly from Corollary 2 and a consequence of representation theory as given in (2 of I in [7 p 9] Therefore (2 equals (20 which completes the proof The obvious utility of Theorem ( 2 is that we only need { to find } the eigenvalues and eigenvectors of the single 2 by 2 matrix instead of those of 3 The Krawtchouk polynomials and matrices β The Krawtchouk polynomials which have been discussed and used extensively (eg [ 69] are a special case of the functions considered here The Krawtchouk orthogonal polynomials are defined (see eg [6] for n x 0by K n (xp n ( x ν ν0 ( x n ν ( p ν ( n β (3 where 0 <p< ote that for n x 0 ( λ nx ( p ; K n n (xp (32 Therefore B p is a Krawtchouk matrix and its inverse is B p p p p The expression (33 extends those found in say [3 5] for the case p /2 ote that in [4] the matrix B evaluated at p /2 is denoted by K ( Thus (33 (34
K ( with inverse [K ( ] C Severo / Linear Algebra and its Applications 429 (2008 50 56 55 /2 /2 /2 /2 Theorem 3 For 0 <p< a spectral representation of the Krawtchouk matrix is given by { w 0 w2 B (w 2 w w w 2 0 w 2 w } where w 4p 2p + 2 + and w 2 4p 2p 2p 2 + 2p Proof Apply Theorem 2 See eg Section II of [3] for a treatment for the case p /2 From Theorem 3 we obtain the following found also in [6]: Corollary The j th eigenvalue and associated eigenvectorj 0for the Krawtchouk matrix are respectivelyw j 2 w j and A(w 2 w ; j A( ; j0 Briefly denote the i jth elements of (33 and (34 byb ij and bij respectively Then we can easily use Theorem to derive identities from expressions such as B B B B b mn B B B B j0 i0 b mi b ij b jn j0 i0 B B For example from the first equality we write whereupon using (32 gives the identity ( K m (n p K m (ipk i (jp λ i jn ( p p p p; As a further illustration of the matrices and applications of Theorem we show that the transition probabilities for the Ehrenfest 2-urn model given by [6] viz ( ( p j ( P ij (t e xt K j i (xpk j (xp p x q x (35 q x x0 for i j 0 is equal to that given by say [8] viz (in our present notation P ij (t λ ji (α β γ ; (36 where α q + ( qe t γ q qe t β p pe t p + ( pe t
56 C Severo / Linear Algebra and its Applications 429 (2008 50 56 ote that since the Krawtchouk polynomials are self-dual ie K n (j K j (n (see Eq (37 of [6] the right-hand side of (35 is equal to ( p j ( ( K q j j (xpe xt p x q x K x x (ip x0 which we write as x0 λ jj ( 0 0 p q This gives rise to 0 p 0 q p b jx λ xx ( 0 0e t λ xx (q 0 0pb xi { 0 0 e } t q 0 0 p p Successively applying Theorem to the latter expression gives { q + ( qe t q qe t p pe t p + ( pe } t Acknowledgments I should like to thank Paul J Schillo for helpful comments at early stages in the development of this work References [] CF Dunkl A Krawtchouk polynomial addition theorem and wreath products of symmetric groups Indiana Univ Math J 25 (976 335 357 [2] GK Eagleson A characterization theorem for positive definite sequences on the Krawtchouk polynomials Austral Z J Statist (969 29 38 [3] P Feinsilver R Fitzgerald The spectrum of symmetric Krawtchouk matrices Linear Algebra Appl 235 (996 2 39 [4] P Feinsilver J Kocik Krawtchouk polynomials and Krawtchouk matrices in: R Baeza-Yates et al (Eds Recent Advances in Applied Probability 2005 Springer-Verlag pp 5 43 <http://apsarxivorg/abs/quant-ph/0702073> [5] P Feinsilver J Kocik Krawtchouk matrices from classical and quantum random walks Contemp Math 287 (200 83 96 <http://arxivorg/abs/quant-ph/070273v> [6] S Karlin J McGregor Ehrenfest urn models J Appl Probab 2 (965 352 376 [7] MA aimark AI Stern Theory of Group Representations Springer-Verlag ewyork Y 982 [8] C Severo A note on the Ehrenfest multiurn model J Appl Probab 7 (970 444 445 [9] G Szego Orthogonal Polynomials Colloquium Publications American Mathematics Society Providence RI 959