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# 7.8: Review Problems

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1. Consider the linear system:

$$\begin{array}{r}x^{1}~~~~~~~~~~~~~~~~~~~~~~~~~~~=v^{1} \\ l^{2}_{1}x^{1}+x^{2}~~~~~~~~~~~~~~~~~~~=v^{2} \\ \vdots~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\vdots\\ l^{n}_{1}x^{1}+l^{n}_{2}x^{2}+\cdots+x^{n}=v^{n}\end{array}$$

$$i.$$ Find $$x^{1}$$.
$$ii.$$ Find $$x^{2}$$.
$$iii.$$ Find $$x^{3}$$.
$$k.$$ Try to find a formula for $$x^{k}$$. Don't worry about simplifying your answer.

2. Let $$M=\begin{pmatrix} X & Y \\ Z & W \end{pmatrix}$$ be a square $$n\times n$$ block matrix with $$W$$ invertible.

$$i.$$ If $$W$$ has $$r$$ rows, what size are $$X$$, $$Y$$, and $$Z$$?

$$ii.$$ Find a $$UDL$$ decomposition for $$M$$. In other words, fill in the stars in the following equation:
$\begin{pmatrix} X & Y \\ Z & W \end{pmatrix}= \begin{pmatrix} I & * \\ 0 & I \end{pmatrix} \begin{pmatrix} * & 0 \\ 0 & * \end{pmatrix} \begin{pmatrix} I & 0 \\ * & I \end{pmatrix}$

3. Show that if $$M$$ is a square matrix which is not invertible then either the matrix matrix $$U$$ or the matrix $$L$$ in the LU-decomposition $$M=LU$$ has a zero on it's diagonal.

4. Describe what upper and lower triangular matrices do to the unit hypercube in their domain.

5. In chapter 3 we saw that since, in general, row exchange matrices are necessary to achieve upper triangular form, $$LDPU$$ factorization is the complete decomposition of an invertible matrix into EROs of various kinds. Suggest a procedure for using $$LDPU$$ decompositions to solve linear systems that generalizes the procedure above.

6. Is there a reason to prefer $LU$ decomposition to $$UL$$ decomposition, or is the order just a convention?

7. If $$M$$ is invertible then what are the $$LU,\,LDU,$$ and $$LDPU$$ decompositions of $$M^{-1}$$ in terms of the decompositions for $$M$$?

8. Argue that if $$M$$ is symmetric then $$L=U^{T}$$ in the $$LDU$$ decomposition of $$M$$.