Practice Problems 6

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Information about Practice Problems 6

Published on March 6, 2014

Author: anhtuantran509


Mathematical Methods for Quantitative Finance Practice Problems 6 1. The full A = QR factorization presented during the lectures contains more information than necessary to reconstruct A. ˜ ˜ ˜˜ (a) What are the smallest matrices Q and R such that QR = A? (b) Use the reduced QR factorization obtained in part (a) to find an expression for the matrix H = A(AT A)−1 AT . How many matrices must be inverted? (note: inverting an orthogonal matrix by taking its transpose doesn’t count) 2. Give an example of a 2 × 2 matrix that has no real eigenvectors. Justify your solution without using any mathematics. 3. Let x and y be vectors of m elements. The least squares solution for a best-fit line for a plot of y versus x is ˆ β = (X T X)−1 X T y where   | | X = 1 x | | (a) Suppose you have a full singular value factorization X = U ΣV T (i.e., the singular ˆ value factorization describe in the lectures). Find an expression for β in terms of U , Σ, and V . Hint: only square matrices are invertible. (b) Like the full QR factorization, the full singular value factorization contains more information than necessary. Find a reduced singular value factorization of X and use it to simplify your answer to part (a). Justify all calculations. ˜ ˜ ˜˜ 4. Let X be an m × n matrix (m > n) whose columns each sum to zero, and let X = QR ˜ be a reduced QR factorization of X. The squared Mahalanobis distance to the point xT ˜i th ˜ is (the i row of X) d2 = xT Σ−1 xi ˜i ˆ ˜ i ˆ where Σ = 1 ˜T ˜ X X m−1 is a covariance matrix. Compute d2 without inverting a matrix. i

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