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# Algorithm to Calculate the Eigenvalues and Eigenvectors of Many Circulant (Cyclic) Matrices

IP.com Disclosure Number: IPCOM000076305D
Original Publication Date: 1972-Feb-01
Included in the Prior Art Database: 2005-Feb-24
Document File: 2 page(s) / 22K

IBM

## Related People

Reynolds, SW: AUTHOR

## Abstract

An algorithm is presented for computing the eigenvalues and eigenvectors for many circulant (cyclic) matrices of the same order. An APL/360 implementation of the algorithm is presented.

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Algorithm to Calculate the Eigenvalues and Eigenvectors of Many Circulant (Cyclic) Matrices

An algorithm is presented for computing the eigenvalues and eigenvectors for many circulant (cyclic) matrices of the same order. An APL/360 implementation of the algorithm is presented.

The algorithm to be presented is novel in that it extends the concept of the eigenvalue and eigenvector calculation for a single matrix to the calculation for many matrices of the same order. The particular matrices under consideration are circulant, or cyclic, matrices.

The principal advantage of this algorithm to an engineer, who recognizes that he is dealing with circulant matrices, is to enable him to compute eigenvalues and eigenvectors for numerous matrices of the same order and to compute them considerably faster than by other methods.

According to an article by Marvin Marcus, Basic Theorems in Matrix Theory, National Bureau of Standards, Applied Mathematics, Series No. 57, page 9 (U.
S. Government Printing Office, Washington, D. C., 1960):

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Both definitions in the article require powers of the n/th/ roots of unity. The second definition clearly shows that the eigenvectors, with complex components, are independent of the elements a(j) of the circulant matrix.

The algorithm to process many circulant matrices will be described, and an APL/360 implementation, called EIGCIRCULANT, will be presented by way of example. This algorithm will work in either 0 or 1 origin indexing. 1) The input (Y) to the monadic fu...