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Exercise 3

Avg difficulty 2.5

Suppose that AA is a demeaned data matrix: A=[1−2011−1−11].A = \begin{bmatrix} 1 & -2 & 0 & 1 \\ 1 & -1 & -1 & 1 \\ \end{bmatrix}.

  1. Find the covariance matrix CC.

  2. What is the variance of the data projected onto the line defined by u=[1/21/2]\uvec=\twovec{1/\sqrt{2}}{1/\sqrt{2}}.

  3. What is the total variance?

  4. In which direction is the variance greatest and what is the variance in this direction?

Austin, Understanding Linear Algebra, §7.2 Quadratic forms, Exercise 3

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Adapted from Understanding Linear Algebra © 2023 David Austin. Licensed under CC BY 4.0. Source: Understanding Linear Algebra. Imported at revision 3e6dced.