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Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.

Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.
Transcribed Image Text:Let X Є pxn denote a zero mean observation matrix matrix of obser- vations, P be a pxp orthogonal matrix, and Y = PTX. (a) Show that if C is the covariance matrix of X then PTCP is the co- variance matrix of Y = PT CP. (b) It can be verified that tr (FG) = tr (GF) for any two n x n matrices F and G. Use this fact to show that the total variance of the data in Y is equal to the total variance of the data in X.
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