Reference no: EM131387522
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 = QΛQT.
(c) Is A positive definite?
2. Principal component analysis: You have 15 observations of elements in R2, v1, . . . , vM, M = 15, vm ∈ R2, which you represent in a 2 × 15 matrix, X ∈ R2×15. The observations are:
Note that ∑m Xnm = 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 xcT, 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||vm - xcm||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?
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