[Solved] CFRM405 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 np 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 mn matrix (assume m > n). The full singular value factorization A = UV ^T contains more information than necessary to reconstruct A.
   • What are the smallest matrices U, and V^T such that UV^T = A?
   • 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 U.

Find expressions for U^TU and UU^T.

• Use the reduced singular value factorization obtained in part (a) to find an expression for the matrix H = A(A^TA)¹A^T. How many matrices must be inverted (diagonal and orthogonal matrices dont count)?

4. Let x and y be vectors of m The least squares solution for a best-fit line for a plot of y versus x is

= (XTX)1XTy

where

• Suppose you know the full singular value factorization X = UV ^T. Find an expression for in terms of U, , and V . Hint: Only square matrices can be invertible.
• Repeat part (a) using the reduced singular value factorization X = UV^T.

5. Let X be an m n matrix (m > n) whose columns have sample mean zero, and let X = UV^T be a reduced singular value factorization of X. The squared Mahalanobis distance to the point x_i^T (the i^th row of X) is

d2i = xiTS1xi

where = cov(X). Explain how to compute d²_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

• 2 1

3 2

• 1 1

by hand using elimination matrices.