Krawtchouk matrices

In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points.^{[1] [2]} The Krawtchouk matrix K^(N) is an (N+1)×(N+1) matrix. Here are the first few examples:

$K^{(0)}=\begin{bmatrix} 1 \end{bmatrix} \qquad K^{(1)}=\left [ \begin{array}{rr} 1&1\\ 1&-1 \end{array}\right ] \qquad K^{(2)}=\left [ \begin{array}{rrr} 1&1&1\\ 2&0&-2\\ 1&-1&1 \end{array}\right ] \qquad K^{(3)}=\left [ \begin{array}{rrrr} 1&1&1&1\\ 3&1&-1&-3\\ 3&-1&-1&3\\ 1&-1&1&-1 \end{array}\right ]$

$K^{(4)}=\left [ \begin{array}{rrrrr} 1&1&1&1&1\\ 4&2&0&-2&-4\\ 6&0&-2&0&6\\ 4&-2&0&2&-4\\ 1&-1&1&-1&1 \end{array}\right ] \qquad K^{(5)}=\left [ \begin{array}{rrrrrr} 1& 1& 1& 1& 1& 1\\ 5& 3& 1&-1&-3&-5\\ 10& 2&-2&-2& 2& 10\\ 10& -2&-2& 2& 2&-10\\ 5& -3& 1& 1&-3&5\\ 1& -1& 1&-1& 1&-1 \end{array}\right ].$

In general, for positive integer $N$, the entries $K^{(N)}_{ij}$ are given via the generating function

$(1+v)^{N-j}\,(1-v)^j=\sum_i v^i K^{(N)}_{ij}$

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, $p=1/2$.^[3]

See also

• Square matrix

References

1. N. Bose, “Digital Filters: Theory and Applications” [North-Holland Elsevier, N.Y., 1985]
2. P. Feinsilver, J. Kocik: Krawtchouk polynomials and Krawtchouk matrices, Recent advances in applied probability, Springer-Verlag, October, 2004
3. Hahn Class: Definitions

External links

• Krawtchouk encyclopedia