Mathematical Methods for Quantitative Finance Homework 5
1. Give an example of a 2 × 2 matrix that has no real eigenvectors. Justify your solution with intuition (without solving completely for the eigenvectors and eigenvalues).
2. Consider an n×p matrix A. Show that the number of linear independent rows is the same as the number of linearly independent columns.
Hint: Write A = CR where C is a matrix of the linearly independent columns of A. Why can we write A like this? Then consider the CR product in the “row” interpretation of matrix multiplication.
3. Let A be an m×n matrix (assume m > n). The full singular value factorization A = UΣV T contains more information than necessary to reconstruct A.
(a) What are the smallest matrices U˜, Σ and˜ V˜T such that U˜Σ˜V˜T = A?
(b) Let . That is, think about U from the full singular value factorization as a block matrix consisting of the matrix U˜ found in part (a) and the remaining (unneeded) columns Uˆ.
Find expressions for U˜TU˜ and U˜U˜T.
(c) Use the reduced singular value factorization obtained in part (a) to find an expression for the matrix H = A(ATA)−1AT. How many matrices must be inverted (diagonal and orthogonal matrices don’t count)?
4. 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
βˆ = (XTX)−1XTy
where
(a) Suppose you know the full singular value factorization X = UΣV T. Find an expression for βˆ in terms of U, Σ, and V . Hint: Only square matrices can be invertible.
(b) Repeat part (a) using the reduced singular value factorization X = U˜Σ˜V˜T.
5. Let X˜ be an m × n matrix (m > n) whose columns have sample mean zero, and let X˜ = U˜Σ˜V˜T be a reduced singular value factorization of X˜. The squared Mahalanobis distance to the point ˜xiT (the ith row of X˜) is
d2i = x˜iTSˆ−1x˜i
where = cov(X˜). Explain how to compute d2i without inverting a matrix.
6. (a) Suppose A = LU where L is lower triangular and U is upper triangular. Explain how you would solve the problem Ax = b using L, U, and the concepts of forward and backward substitution.
(b) Compute the LU factorization of
1 2 −1
−3 2
2 1 1
by hand using elimination matrices.
