Exercise 4

Determine whether the following statements are true or false and explain your thinking.

1. If the eigenvalues of the covariance matrix are $\lambda_1$, $\lambda_2$, and $\lambda_3$, then $\lambda_3$ is the variance of the demeaned data points when projected on the third principal component $\uvec_3$.

2. Principal component analysis always allows us to construct a smaller dimensional representation of a dataset without losing any information.

3. If the eigenvalues of the covariance matrix are 56, 32, and 0, then the demeaned data points all lie on a line in $\real^3$.