Wa3_sol - Winter 2011 Math 67 Linear Algebra Writing Assignment 3 Exercise 1 The circulant matrix x1 x4 X= x3 x2 x2 x1 x4 x3 x3 x2 x1 x4 x4 x3 C44

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Winter 2011 Math 67 Linear Algebra Writing Assignment 3 Exercise 1. The circulant matrix X = x 1 x 2 x 3 x 4 x 4 x 1 x 2 x 3 x 3 x 4 x 1 x 2 x 2 x 3 x 4 x 1 C 4 × 4 has an eigenvector (1 ,ζ,ζ 2 3 ), where ζ is a 4th root of unity. The associated eigen- value is x 1 + ζx 2 + ζ 2 x 3 + ζ 3 x 4 . Proof. Observe that the following system of equations is consistent for λ = x 1 + ζx 2 + ζ 2 x 3 + ζ 3 x 4 , given that ζ 4 = 1. x 1 x 2 x 3 x 4 x 4 x 1 x 2 x 3 x 3 x 4 x 1 x 2 x 2 x 3 x 4 x 1 1 ζ ζ 2 ζ 3 = x 1 + ζx 2 + ζ 2 x 3 + ζ 3 x 4 x 4 + ζx 1 + ζ 2 x 2 + ζ 3 x 3 x 3 + ζx 4 + ζ 2 x 1 + ζ 3 x 2 x 2 + ζx 3 + ζ 2 x 4 + ζ 3 x 1 = λ λζ λζ 2 λζ 3 ± Exercise 2. The matrix Z = 1 1 0 0 0 1 1 0 0 0 1 1 1 0 0 1 R 4 × 4 has eigenvalues 0,1 - i ,1 + i , and 2. The eigenvectors of a general 4 × 4 circulant matrix are linearly independent. Proof. Since Z is a circulant matrix, it follows from the result of problem 1 that it has an eigenvector (1 ,ζ,ζ 2 3 ) with associated eigenvalue 1+
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This note was uploaded on 11/13/2011 for the course MATH 67 taught by Professor Schilling during the Winter '07 term at UC Davis.

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