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Mathematics LibreTexts

7.8: Review Problems

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 \(M\). 

 

 

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