Menu
Home
People
Places
Arts
History
Plants & Animals
Science
Life & Culture
Technology
Reference.org
Krawtchouk matrices
open-in-new
<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Krawtchouk matrices are <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrices</a> whose entries are values of <a href="/facts/Krawtchouk_polynomials/Uh7LlP0h">Krawtchouk polynomials</a> at nonnegative integer points. The Krawtchouk matrix <i>K</i>(<i>N</i>) is an (<i>N</i> + 1) × (<i>N</i> + 1) matrix. The first few Krawtchouk matrices are: </p> 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 ] , {\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],} 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 ] . {\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].}