The 1, -1 first difference matrix A has AT A = second difference matrix. The singular

Chapter 6, Problem 17

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The 1, -1 first difference matrix A has AT A = second difference matrix. The singular vectors of A are sine vectors v and cosine vectors u. Then Av = (JU is the discrete form of d/dx(sincx) = c(coscx). This is the best SVD I have seen. 1 0 0 SYDor A A= -1 I 0 o -1 1 o 0-1 ATA= -1 2-1 [ 2 -1 0] Orthogonal sine matrix 1 [Sinn/4 V = - sin2n/4 ,Ji sin 3n / 4 o -1 2 sin 2n / 4 sin 3n I 4] sin4n/4 sin6nl4 sin 6n / 4 sin 9n / 4 (a) Put numbers in V: The unit eigenvectors of AT A are singular vectors of A. Show that the columns of V have AT Av = AV with A = 2 - ,Ji, 2, 2 + ,Ji. (b) Multiply AV and verify that its columns are orthogonal. They are (JlUl and (J2U2 and (J3U3. The first columns of the cosine matrix U are Ul, U2, U3. (c) Since A is 4 by 3, we need a fourth orthogonal vector U4. It comes from the nullspace of AT. What is U4? The cosine vectors in U are eigenvectors of AAT. The fourth cosine is (1, 1, 1, 1)/2. 1 -1 0 0 cos n 18 cos 2n /8 cos 3 n / 8 AAT = -1 2 -IOU = _1_ cos3n/8 cos6n/8 cos9n/8 o -1 2 -1 ,Ji cos5n/8 cos lOnl8 cos 15nl8 o 0 -1 1 cos7n/8 cos 14n/8 cos21n/8 Those angles n /8, 3n 18, 5n /8, 7n /8 fit 4 points with spacing n / 4 between 0 and n. The sine transform has three points n / 4, 2n / 4, 3n / 4. The full cosine transform includes U4 from the "zero frequency" or direct current eigenvector (1, 1, 1, 1). The 8 by 8 cosine transform in 2D is the workhorse of jpeg compression. Linear algebra (circulant, Toeplitz, orthogonal matrices) is at the heart of signal processing.