Raising the bar

Questions for repetition: projectors and applications

1. If a matrix is idempotent, then it can have only two eigenvalues: 0 and 1.

2. If a matrix is idempotent, then its image is the set of all vectors that are unchanged by this matrix (such vectors are called fixed points).

3. The null space and the image of a projector are orthogonal.

4. If $latex P$ is a projector, then $latex Q=I-P$ is too. What is the relationship between the image and null space of $latex P$ and those of $latex Q?$

5. Using a basis in a subspace $latex L,$ how do you construct a projector $latex P$ onto $latex L?$ Prove existence, idempotency and symmetry. Prove that $latex \text{Img}(P)=L$.

6. Prove that if $latex P$ is the projector from the previous exercise, then $latex Q=I-P$ projects onto $latex L^\perp.$ Thus, for a given $latex L,$ we have a way to find $latex L^\perp.$ Use this fact to prove the theorem on the second orthocomplement.

7. Prove the generalized Pythagoras theorem.

8. Define the OLS estimator for the linear model and derive it. Why does the condition of linear independence of regressors imply that $latex k\leq n?$

9. How does the simplified derivation of the OLS estimator look like? Why is it not rigorous?

10. Find eigenvalues of a projector. In particular, prove that eigenvectors corresponding to $latex \lambda=1$ are orthogonal to eigenvectors corresponding to $latex \lambda=0.$

11. Prove the trace-commuting property.

12. What is the relationship between the trace and rank of a projector?

13. Using $latex P=X(X^TX)^{-1}X^T$, find the expressions for the fitted value and residual in the linear model theory.

14. Define the OLS estimator of $latex \sigma^2$ and prove its unbiasedness. Note the crucial role of the assumption $latex Var(e)=\sigma^2I.$ Without it there is no unbiasedness and Gauss-Markov theorem.

15. How do you define a standard normal vector? What are its mean and variance?

16. How do you define a general normal vector? Find its mean and variance.

17. Find the mean and variance of $latex \chi_n^2.$ Prove that $latex \chi_n^2/n$ converges in probability to 1.

18. Characterize the distribution of the OLS estimator of $latex \sigma ^{2}.$ Use it to find its mean and variance.

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