Introduction to Linear Algebra

Systems of Linear Equations

The inverse of block matrix \( \displaystyle \quad \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0} \\ \mathbf{A} & \mathbf{I} & \mathbf{0} \\ \mathbf{B} & \mathbf{C} & \mathbf{I} \end{bmatrix} \quad \) is \( \displaystyle \quad \begin{bmatrix} \mathbf{I} & \mathbf{0} & \mathbf{0} \\ \mathbf{X} & \mathbf{I} & \mathbf{0} \\ \mathbf{Y} & \mathbf{Z} & \mathbf{I} \end{bmatrix} . \quad \) Find matrices X, Y, and Z.
Let X be an m × n data matrix such that X^TX is invertible, and let M = I_m − X(X^TX)⁻¹X^T. Add a column x₀ to the data and form the augmented matrix \[ {\bf W} = \begin{bmatrix} \mathbf{X} & \mathbf{x}_0 \end{bmatrix} . \] Compute W^TW. The (1, 1)-entry is X^TX. Show that the Schur complement of X^TX can be written in the form x₀^T M x₀. It can be shown that the quantity (x₀^T M x₀)⁻¹ is the (2, 2)-entry in W^TW. This entry has a useful statistical interpretation, under appropriate hypotheses.

Determinants Cofactors Cramer's Rule Partitioned Matrices Elementary Matrices Inverse Matrices Elimination: A = LU

Without row exchange, use elementary matrices to find LU-factorizations for the following matrices. \[ \mbox{(a)} \quad \begin{bmatrix} -8&-5&-6&5 \\ 3&-6&7&-3 \\ -10&-3&4&2 \\ 5&-5&7&8 \end{bmatrix} ; \qquad \mbox{(b)} \quad \begin{bmatrix} 2&1&-1&0 \\ 4&3&3&1 \\ 8&7&9&5 \\ 6&7&9&8 \end{bmatrix} ; \qquad \mbox{(c)} \quad \]

PLU Factorization

Using row exchange and elementary matrices, find PLU-factorizations for the following matrices.
1. \[ \mbox{(a)} \quad \begin{bmatrix} -9&8&-3&0 \\ -9&-5&5&1 \\ 6&7&3&5 \\ 6&-2&6&7 \end{bmatrix} , \qquad \mbox{(b)} \quad \begin{bmatrix} 5&2&-5&0 \\ 6&-8&-8&-4 \\ 2&9&-9&2 \\ -1&6&6&-6 \end{bmatrix} \]
Reflection Givens Rotation Special Matrices