Introduction to the Theory of Computation (Solution Manual ) by Law C., Lee E.K., Ramzan Z.,Michael Sipser

By Law C., Lee E.K., Ramzan Z.,Michael Sipser

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A = (Aj−i mod n )i,j=1,n =   .  .. A   .. 1 A1 . . An−1 A0 is called the block circulant matrix associated with [A0 , A1 , . . , An−1 ] and is denoted by Circ(A0 , A1 , , . . , An−1 ). 6 If A is a block circulant matrix with first block row r T and with first block column c we have A= 1 (Ωn ⊗ Im ) Diag(W1 , . . , Wn )(Ωn ⊗ Im ) n where [W1 , . . , Wn ] = r T (Ωn ⊗ Im ),   W1  ..   .  = (Ωn ⊗ Im )c. Wn 30 STRUCTURED MATRIX ANALYSIS Like circulant matrices, the class of block-circulant matrices is closed under matrix multiplication and inversion.

9, we may assume that A is in the following form   0 A1,1   A2,1 A2,2   A= .  . ..   .. Ak,1 . . Ak,k−1 Ak,k where Ai,i , i = 1, . . , k are irreducible. Since A1,1 corresponds to the final class, then ρ(A) = ρ(A1,1 ). ,k . We prove the theorem by induction on the number of irreducible classes k. For k = 1 the matrix A is irreducible so that for the Perron–Frobenius theorem x > 0. Assume that the theorem holds for k − 1 irreducible classes and let us prove it for k irreducible classes.

A1 ]T . Any other row or column is obtained from the preceding one by applying a cyclic permutation to its elements: the last element is moved to the first position and the remaining ones are shifted by one position. With C denoting the circulant matrix associated with [0, 1, 0, . . ,   0 1 0 ... 0  .   0 0 1 . . 5) C =  ... . . . . 0  ,     .  0 .. 0 1  1 0 ... 6) i=0 that is, any circulant matrix can be viewed as a polynomial in C. By direct inspection we see that CΩn = Ωn Diag(1, ω n , ω 2n , .

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