Mfe spring 2017 mathematical foundations assignment

Question

Mathematical Foundations Assignment

In this assignment you will practice eigenvector decompositions, principal component analysis and stochastic matrix theory.

1. Eigenvector decomposition: Consider the matrix

(a) What are A's eigenvectors and eigenvalues?

(b) What are the matrices Q and Λ in the spectral decomposition A = QΛQ^T.

(c) Is A positive definite?

2. Principal component analysis: You have 15 observations of elements in R², v¹, . . . , v^M, M = 15, v^m ∈ R², which you represent in a 2 × 15 matrix, X ∈ R^2×15. The observations are:

Note that ∑_m X_nm = 0, n = 1, 2.  You wish to explore whether the dataset is well represented by elements in a subspace, E ⊂ R².  Specifically, you want to know how well you can represent X by xc^T, where x ∈ R², and c ∈ R¹⁵ (i.e., the subspace is E = span(x)).

Do a principal component analysis to explore how well such a sub-space representation works in this case.

(a) What is the optimal x?

(b) What is the error e = 1/M√(∑_m||v^m - xc_m||²₂)?

(c) In light of your previous results, is it possible to represent the v's well in a 1-dimensional subspace, E?

3. Stochastic matrices: Consider the matrix:

(a) Show that Φ is a stochastic matrix.

(b) Is Φ aperiodic?

(c) Is Φ irreducible?

(d) Does Φ have a unique stationary distribution?