In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points. The Krawtchouk matrix K(N) is an (N+1)×(N+1) matrix. Here are the first few examples:
K ( 0 ) = [ 1 ] K ( 1 ) = [ 1 1 1 − 1 ] K ( 2 ) = [ 1 1 1 2 0 − 2 1 − 1 1 ] K ( 3 ) = [ 1 1 1 1 3 1 − 1 − 3 3 − 1 − 1 3 1 − 1 1 − 1 ]
K ( 4 ) = [ 1 1 1 1 1 4 2 0 − 2 − 4 6 0 − 2 0 6 4 − 2 0 2 − 4 1 − 1 1 − 1 1 ] K ( 5 ) = [ 1 1 1 1 1 1 5 3 1 − 1 − 3 − 5 10 2 − 2 − 2 2 10 10 − 2 − 2 2 2 − 10 5 − 3 1 1 − 3 5 1 − 1 1 − 1 1 − 1 ] .
K ( 0 ) = [ 1 ] K ( 1 ) = [ 1 1 1 − 1 ] K ( 2 ) = [ 1 1 1 2 0 − 2 1 − 1 1 ] K ( 3 ) = [ 1 1 1 1 3 1 − 1 − 3 3 − 1 − 1 3 1 − 1 1 − 1 ]
K ( 4 ) = [ 1 1 1 1 1 4 2 0 − 2 − 4 6 0 − 2 0 6 4 − 2 0 2 − 4 1 − 1 1 − 1 1 ] K ( 5 ) = [ 1 1 1 1 1 1 5 3 1 − 1 − 3 − 5 10 2 − 2 − 2 2 10 10 − 2 − 2 2 2 − 10 5 − 3 1 1 − 3 5 1 − 1 1 − 1 1 − 1 ] .
In general, for positive integer N , the entries K i j ( N ) are given via the generating function
( 1 + v ) N − j ( 1 − v ) j = ∑ i v i K i j ( N ) where the row and column indices i and j run from 0 to N .
These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, p = 1 / 2 .
