Does this matrix come up anywhere?

[YaroslavVB]

d-by-d matrix where d is a power of 2

d,1,1,1,...
1,d,1,1,...
1,1,d,1,..
...

In particular, I'm looking for nice expression for an orthogonal basis of eigenvectors of it

 

[Office_Shredder]

You can decompose it into (d-1)I-A where A is the matrix with all 1's. The eigenvectors of A and (d-1)I-A are the same, so all you need to do is find the eigenvectors of the matrix with all ones.

Since A has a big kernel and you get to pick whichever eigenvectors you want this should be doable

 

[Simon_Tyler]

+1 for office_shredder.

and in answer to the question in your title - this type of matrix does come up in some physical systems... I just can't remember where I've seen it.

Also, it's a very special type of http://en.wikipedia.org/wiki/Circulant_matrix