- 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).
- 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.

- Let
*A*be an*m*×*n*matrix (assume*m > n*). The full singular value factorization*A*=*U*Σ*V*^{T }contains more information than necessary to reconstruct*A*.- What are the smallest matrices
*U*^{˜}, Σ and^{˜ }*V*^{˜}^{T }such that*U*^{˜}Σ^{˜}*V*^{˜}^{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*ˆ.

- What are the smallest matrices

Find expressions for *U*^{˜}^{T}*U*^{˜ }and *U*^{˜}*U*^{˜}^{T}.

- Use the
*reduced*singular value factorization obtained in part (a) to find an expression for the matrix*H*=*A*(*A*^{T}*A*)^{−1}*A*^{T}. How many matrices must be inverted (diagonal and orthogonal matrices don’t count)?

- 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

*β*ˆ = (*X*T*X*)−1*X*T*y*

where

- Suppose you know the
**full**singular value factorization*X*=*U*Σ*V*^{T}. Find an expression for*β*^{ˆ }in terms of*U*, Σ, and*V*. Hint: Only square matrices can be invertible. - Repeat part (a) using the reduced singular value factorization
*X*=*U*^{˜}Σ^{˜}*V*^{˜}^{T}.

- 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*_{i}^{T }(the*i*^{th }row of*X*^{˜}) is

*d*2*i *= *x*˜*i*T*S*ˆ−1*x*˜*i*

where = cov(*X*^{˜}). Explain how to compute *d*^{2}* _{i }*without inverting a matrix.

- (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.

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