Does this matrix come up anywhere?

  • Thread starter YaroslavVB
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In summary, a d-by-d matrix where d is a power of 2 can be decomposed into (d-1)I-A, where A is a matrix with all 1's. The eigenvectors of A and (d-1)I-A are the same, so finding the eigenvectors of the matrix with all ones is all that is needed. This type of matrix is known as a circulant matrix and is commonly used in physical systems.
  • #1
YaroslavVB
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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
 
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  • #2
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
 
  • #3
+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
 

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A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is important in a variety of fields, including computer science, physics, and engineering, as it provides a convenient way to organize and manipulate data.

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In computer science, matrices are commonly used to represent and manipulate data, such as images or text. They are also used in algorithms and machine learning techniques to solve problems and make predictions.

3. Can you give an example of where matrices are used in physics?

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