Mathematical Methods for Quantitative Finance Homework 5

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 U =

U˜ Uˆ

. 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 = (X
TX)
−1X
T
y
where
X =


| |
1 x
| |


(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 ˜x
T
i
(the i
th row of X˜) is
d
2
i = ˜x
T
i Sˆ−1x˜i
where Sˆ =
1
m−1X˜ TX˜ = cov(X˜). Explain how to compute d
2
i 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
A =


1 2 −1
3 −3 2
−2 1 1


by hand using elimination matrices.
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