# Does this kind of matrix have a name?

Are these kind of matrices generally known in mathematics? Do they have a name?

$$\left[\begin{array}{rrr} A & B \\ B & A \\ \end{array}\right]$$

$$\left[\begin{array}{rrr} A & B & C \\ C & A & B \\ B & C & A \\ \end{array}\right]$$

$$\left[\begin{array}{rrr} A & B & C & D \\ D & A & B & C \\ C & D & A & B \\ B & C & D & A \\ \end{array}\right]$$

$$\left[\begin{array}{rrr} A & B & C & D & E \\ E & A & B & C & D \\ D & E & A & B & C \\ C & D & E & A & B \\ B & C & D & E & A \\ \end{array}\right]$$

The main thing is that each letter will be in the same columnn/row just once. I'm trying to do some combination calculations with big matrices following this pattern, so knowing effective ways to generate and compute these would help.

(The pattern here is that the next row is made by shifting the previous row to one right.)

• The letters A,B,C,D,E represent block submatrices or elements. Is by any chance circulant matrix the name you're looking for? – Martin Sleziak Jul 17 '12 at 15:50
• My first guess would be Toeplitz matrices, though these seem to have some extra structure that looks common enough to have a name attached. You see these sometimes in physics problems, (typically sparse), though I don't know a name per se (which is why I made this a comment). – Robert Mastragostino Jul 17 '12 at 15:59
• Here is a reference on circulant matrices: amazon.com/Circulant-Matrices-AMS-Chelsea-Publication/dp/… – ncmathsadist Jul 17 '12 at 16:10
• If the main idea is that each letter appears in each row and each column exactly once, then there are many examples of that, that do not follow the simple pattern that your examples follow. For example the multiplication tables of finite groups. – Michael Hardy Jul 17 '12 at 17:36