CS3331: Numerical Methods
Homework 2
Due date: March. 25, 1998 (Tuesday)

1.

(10%) Use the method of Gaussian elimination to solve, by hand, the following matrix equation:

$$\left[ \begin{array} {cccc} 2 & 4 & -1 & -2\\ 0 & 1 & -1 ... ...[ \begin{array} {c} -1\\ -5\\ 6\\ 7 \end{array} \right]$$

.

2.

(10%) Find LU decomposition, by hand, of the following matrix:

$$\left[ \begin{array} {ccc} -1 & 2 & 0\\ 2 & -1 & 2\\ 0 & 2 & -1\\ \end{array} \right]$$

.

3.

(10%) Find the inverse, by hand, of the following matrix:

$$\left[ \begin{array} {ccccc} 0 & 0 & 0 & 0 & 2\\ 0 & 2 & ... ...2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ \end{array} \right]$$

. (Note that this is two times a permutation matrix.)

4.

(20%) A simple projection matrix ${\bf P}$ can be expressed as ${\bf a}{\bf a}^T$, where ${\bf a}$ is a unit vector.

(a)

Prove that a simple projection matrix ${\bf P}$ satisfies

$${\bf P}^2 = {\bf P}.$$

(b)

Prove that the trace (sum of diagonal elements) of a simple projection matrix is always 1.

5.

(20%) A more general definition of a projection matrix ${\bf P}$ is a square matrix that satisfies ${\bf P}^2 = {\bf P}$. Use this definition to prove the following two statements:

(a)

If ${\bf P}_1$ and ${\bf P}_2$ are projection matrices, then ${\bf P}_1+{\bf P}_2$ is a projection matrix only if ${\bf P}_1{\bf P}_2+{\bf P}_2{\bf P}_1$ is a zero matrix.

(b)

If ${\bf P}$ is a projection matrix, then ${\bf I}-{\bf P}$ is also a projection matrix.

6.

(30%) Suppose that an invertible matrix ${\bf M}$ is partitioned into four submatrices ${\bf M}= \left[ \begin{array} {cc} {\bf A}& {\bf B}\\ {\bf C}& {\bf D} \end{array} \right],$ where ${\bf A}$ and ${\bf D}$ are also invertible.

(a)

Find the inverse of ${\bf M}$ in terms of the submatrices ${\bf A}$, ${\bf B}$, ${\bf C}$, and ${\bf D}$. Specifically, you should start with the identity $\left[ \begin{array} {cc} {\bf A}& {\bf B}\\ {\bf C}& {\bf D} \end{array}... ...egin{array} {cc} {\bf I}& {\bf 0} \\ {\bf 0} & {\bf I} \end{array} \right]$ and expand it into four matrix equations. Then derive ${\bf X}$, ${\bf Y}$, ${\bf Z}$, and ${\bf U}$ in terms of the submatrices ${\bf A}$, ${\bf B}$, ${\bf C}$, and ${\bf D}$ explicitly.

(b)

Repeat (a), but start with $\left[ \begin{array} {cc} {\bf X}& {\bf Y}\\ {\bf Z}& {\bf U} \end{array}... ...gin{array} {cc} {\bf I}& {\bf 0} \\ {\bf 0} & {\bf I} \end{array} \right].$

(c)

Do you obtain the same answer in (a) and (b)? If yes, prove that the obtained answers are equivalent. If no, explain why.

Note that:

• ${\bf B}$ and ${\bf C}$ are not necessarily invertible. So your answers should not include expressions such as ${{\bf B}}^{-1}$ and ${{\bf C}}^{-1}$.
• In deriving the answers, you should be extremely careful about matrix manipulation techniques. If you have any questions, feel free to consult you Linear Algetra textbooks or ask TA.