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Orthonormal Matrix Transforms

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  • Matrix
  • Orthonormal Matrix

Abstract

Connexions module: m10954 1 Orthonormal Matrix Transforms ∗ Ivan Selesnick This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License † Consider the transformation of the data vector x: y = Qx  y0 y1 . . . yN−1  =  q (0, 0) q (0, 1) . . . q (0, N − 1) q (1, 0) q (1, 1) . . . q (1, N − 1) . . . . . . . . . . . . q (N − 1, 0) q (N − 1, 1) . . . q (N − 1, N − 1)   x0 x1 . . . xN−1  y (k) = ∑ n q (k, n)x (n) If Q is an orthogonal matrix (meaning that QTQ = I) then the data vector x can be recovered from y using the transpose of Q: x = QT y  x0 x1 . . . xN−1  =  q (0, 0) q (1, 0) . . . q (N − 1, 0) q (0, 1) q (1, 1) . . . q (N − 1, 1) . . . . . . . . . . . . q (0, N − 1) q (1, N − 1) . . . q (N − 1, N − 1)   y0 y1 . . . yN−1  x (n) = ∑ k q (k, n) y (k) ∗ Version 2.3: Aug 12, 2005 2:12 pm GMT-5 † http://creativecommons.org/licenses/by/1.0 http://cnx.org/content/m10954/2.3/

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