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Let $\mathbf{X}$ denote a vector that varies over the columns of a $p \times N$ matrix of observations, and let $P$ be a $p \times p$ orthogonal matrix. Show that the change of variable $\mathbf{X}=P \mathbf{Y}$ does not change the total variance of the data. [Hint: By Exercise 11 , it suffices to show that tr $\left(P^{T} S P\right)=\operatorname{tr}(S)$ . Use a property of the trace mentioned in Exercise 25 in Section 5.4 .]

By Exercise 11 , the change of variables $X = P Y$ changes the covariance matrix $S$ of $X$ into the covarinace matrix $P ^ { T } S P$ of $Y$ . The total varince of the data as described by $Y$ is tr $\left( P ^ { T } S P \right)$ . However, since $P T S P$ is similar to $\text { S, they have the same trace (by Exercise } 25 \text { in Section } 5.4 )$ . Thus the total variance of the data is unchanged by the change of variables $X = P Y$ .

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 5

Applications to Image Processing and Statistics

Introduction to Matrices

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