A community for students.
Here's the question you clicked on:
 0 viewing
gabriel2058
 3 years ago
What is a cyclic matrix?
gabriel2058
 3 years ago
What is a cyclic matrix?

This Question is Closed

gabriel2058
 3 years ago
Best ResponseYou've already chosen the best response.0An another matrix named centered matrix is the inverse of cyclic matrix ?

hewsmike
 3 years ago
Best ResponseYou've already chosen the best response.0I assume you are referring to cyclic permutation matrices. These permute the rows or columns of a given matrix ( depending on whether they multiply on the left or on the right ) such that all rows/columns are permuted and you will get back to the original matrix after some number of permutations. Permutations simply means rearrange objects amongst themselves and that can lead to many patterns of use. Taking matrices of size 2 x 2 then \[\left[\begin{matrix}1& 0 \\ 0 & 1\end{matrix}\right]\left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]= \left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]=\left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]\left[\begin{matrix}1& 0 \\ 0 & 1\end{matrix}\right]\]has the identity matrix producing a 'null' permutation. While this seems trivial there is an important point here ( see later ). Now \[\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]\left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]= \left[\begin{matrix}3& 4 \\ 1 & 2\end{matrix}\right]\]and \[\left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]= \left[\begin{matrix}2& 1 \\ 4 & 3\end{matrix}\right]\]emphasising that what a given matrix does under multiplication depends on ordering ie. matrix multiplication does not generally permute. Here a permutation matrix applied to the left side rearranges rows, and if applied to the right side it rearranges columns. So if we apply those permutations again : \[\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]\left[\begin{matrix}3& 4 \\ 1 & 2\end{matrix}\right]= \left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]\]and \[\left[\begin{matrix}2& 1 \\ 4 & 3\end{matrix}\right]\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]= \left[\begin{matrix}1& 2 \\ 3 & 4\end{matrix}\right]\]we arrive back where we started. In fact \[\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]\left[\begin{matrix}0& 1 \\ 1 & 0\end{matrix}\right]= \left[\begin{matrix}1& 0 \\ 0 & 1\end{matrix}\right]\]or \[P^{2}=I\] from which we deduce that \[P=P^{1}\]You'll note that I produced P by rearranging the rows/columns of the identity. We've just exhausted the ways of doing such permutations for 2 x 2, and for 3 x 3 there are more ways. In any case for given dimensions the permutation matrices are what is formally called a 'group'. Any two members of the group when combined from another group member, there is an identity member, and each member has an inverse. My understanding of the term 'centered matrix' doesn't seem to apply here ( please advise ).
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.