Methods of Applied Mathematics Lecture Notes

1.1. MATRICES 15where D is diagonal.Notice that this is the eigenvalue equation AU = UD. The columns of Uare the eigenvectors, and the diagonal entries of D are the eigenvalues.Example: LetP = √ 1 [ ]1 −1. (1.42)2 1 1Then P is a rotation by π/4. The eigenvalues are on the diagonal ofF = √ 1 [ ]1 + i 0. (1.43)2 0 1 − iA suitable unitary matrix Q isThen P Q = QF .Q = √ 1 [12 −i1.1.8 Circulant (convolution) matrices]i. (1.44)−1If A is normal, then there are unitary U and diagonal D with AU = UD. Butthey are difficult to compute. Here is one special but important situation whereeverything is explicit.A circulant (convolution) matrix is an n by n matrix such that there is afunction a with A pq = a(p − q), where the difference is computed modulo n.[For instance, a 4 by 4 matrix would have the same entry a(3) in the 12, 23, 34,and 41 positions.]The DFT (discrete Fourier transform) matrix is an n by n unitary matrixgiven byU qk = √ 1 e 2πiqkn . (1.45)nTheorem. Let A be a circulant matrix. If U is the DFT matrix, thenwhere D is a diagonal matrix withU −1 AU = D, (1.46)D kk = â(k) = ∑ ra(r)e − 2πirkn . (1.47)1.1.9 Problems1. Let⎡⎤5 1 4 2 31 2 6 3 1A = ⎢ 4 6 3 0 4 ⎥⎣⎦ . (1.48)2 3 0 1 23 1 4 2 3Use Matlab to find orthogonal P and diagonal D so that P −1 AP = D.