The eigenvectors and eigenvalues of block circulant matrices had been found
for real symmetric matrices with symmetric submatrices, and for block circulant
matrices with circulant submatrices. The eigenvectors are now found
for general block circulant matrices, including the Jordan Canonical Form
for defective eigenvectors. That analysis is applied to Stephen J. Watson’s
alternating circulant matrices, which reduce to block circulant matrices with
square submatrices of order 2

Tee, Garry J.

English

Massey Research Online

Eigenvectors of block circulant and alternating circulant matrices

Abstract. The eigenvectors and eigenvalues of symmetric block circulant ma-trices had been found, and that method is extended to general block circulant matrices. That analysis is applied to Stephen J. Watson’s alternating circulant matrices, which reduce to block circulant matrices with square submatrices of order 2. 1. Circulant Matrix A square matrix in which each row (after the first) has the elements of the previous row shifted cyclically one place right, is called a circulant matrix. Philip R. Davis (1979, p.69) denotes it as B = circ(b0, b1,  ·  ·  · , bn−1) def= b0 b1 b2... bn−2 bn−1 bn−1 b0 b1... bn−3 bn−2 bn−2 bn−1 b0... bn−4 bn−3 b2 b3 b4... b0 b1 b1 b2 b3... bn−1 b

Garry J. Tee

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