Kravchuk Matrix
Get Kravchuk Matrix essential facts below. View Videos or join the Kravchuk Matrix discussion. Add Kravchuk Matrix to your Like2do.com topic list for future reference or share this resource on social media.
Kravchuk Matrix

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:

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

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

${\displaystyle (1+v)^{N-j}\,(1-v)^{j}=\sum _{i}v^{i}K_{ij}^{(N)}}$

where the row and column indices ${\displaystyle i}$ and ${\displaystyle j}$ run from ${\displaystyle 0}$ to ${\displaystyle N}$.

These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, ${\displaystyle p=1/2}$.[3]

## References

1. ^ Bose, N. (1985). Digital Filters: Theory and Applications. New York: North-Holland Elsevier. ISBN 0-444-00980-9.
2. ^ Feinsilver, P.; Kocik, J. (2004). Krawtchouk polynomials and Krawtchouk matrices. Recent Advances in Applied Probability. Springer-Verlag. arXiv:quant-ph/0702073. Bibcode:2007quant.ph..2073F.
3. ^ "Hahn Class: Definitions". Digital Library of Mathematical Functions.