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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:
![\begin{displaymath}
\left[
\begin{array}
{cccc}
2 & 4 & -1 & -2\\ 0 & 1 & -1 ...
...[
\begin{array}
{c}
-1\\ -5\\ 6\\ 7
\end{array} \right]
\end{displaymath}](img1.gif)
.
- 2.
- (10%) Find LU decomposition, by hand, of the following matrix:
![\begin{displaymath}
\left[
\begin{array}
{ccc}
-1 & 2 & 0\\ 2 & -1 & 2\\ 0 & 2 & -1\\ \end{array} \right]
\end{displaymath}](img2.gif)
.
- 3.
- (10%) Find the inverse, by hand, of the following matrix:
![\begin{displaymath}
\left[
\begin{array}
{ccccc}
0 & 0 & 0 & 0 & 2\\ 0 & 2 & ...
...2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ \end{array} \right]
\end{displaymath}](img3.gif)
.
(Note that this is two times a permutation matrix.)
- 4.
- (20%) A simple projection matrix
can be expressed as
, where
is a unit vector.
- (a)
- Prove that a simple projection matrix
satisfies

- (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
is
a square matrix that satisfies
. Use this definition to prove the following two statements:
- (a)
- If
and
are projection matrices,
then
is a projection matrix only if
is a zero matrix.
- (b)
- If
is a projection matrix, then
is also a
projection matrix.
- 6.
- (30%) Suppose that an invertible matrix
is partitioned
into four submatrices
where
and
are also invertible.
- (a)
- Find the inverse of
in terms of
the submatrices
,
,
, and
. Specifically, you should start with the identity
and expand it into four matrix equations.
Then derive
,
,
, and
in terms of the
submatrices
,
,
, and
explicitly.
- (b)
- Repeat (a), but start with
- (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:
-
and
are not necessarily invertible.
So your answers should not include expressions such as
and
. - 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.
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J.-S. Roger Jang
3/18/1998