A persymmetric matrix is a matrix that is symmetric about both diagonals; that is, an N × N matrix A = (aij) is persymmetric if aij = aji = aN+1−i,N+1−j, for all i = 1, 2, . . . , N and j = 1, 2, . . . , N. A number of problems in communication theory have solutions that involve the eigenvalues and eigenvectors of matrices that are in persymmetric form. For example, the eigenvector corresponding to the minimal eigenvalue of the 4 × 4 persymmetric matrix
gives the unit energy-channel impulse response for a given error sequence of length 2, and subsequently the minimum weight of any possible error sequence.
a. Use the Geršgorin Circle Theorem to show that if A is the matrix given above and λ is its minimal eigenvalue, then |λ − 4| = ρ(A − 4I), where ρ denotes the spectral radius.
b. Find the minimal eigenvalue of the matrix A by finding all the eigenvalues A−4I and computing its spectral radius. Then find the corresponding eigenvector.
c. Use the Geršgorin Circle Theorem to show that if λ is the minimal eigenvalue of the matrix
then |λ − 6| = ρ(B − 6I).
d. Repeat part (b) using the matrix B and the result in part (c).
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