The LU-Factorization: Homework

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Section 3.15 The LU-Factorization: Homework

Exercises Exercises

Exercise Group.

Find the \(LU\)-factorization of the matrix \(\A\) in both ways: first by Algorithm 1, then by Algorithm 2. Then find the \(LD\widetilde{U}\)-factorization of \(\A\text{.}\)

1.

\(\A=\left(\begin{array}{rrr}2\amp -2\amp 4\\-6\amp 11\amp 3\\8\amp -33\amp -65 \end{array} \right)\)

2.

\(\A=\left(\begin{array}{rrr}1\amp -2\amp -4\\2\amp -3\amp -9\\-5\amp 16\amp 15 \end{array} \right)\)

Exercise Group.

Find the \(LU\)-factorization of \(\A\text{.}\)

3.

\(\A=\left(\begin{array}{rrr}3\amp 1\amp -5\\12\amp 8\amp -22\\6\amp -26\amp 16 \end{array} \right)\)

4.

\(\A=\left(\begin{array}{rrr}1\amp 2\amp -4\\2\amp -3\amp -9\\-4\amp -9\amp 15 \end{array} \right)\)

Exercise Group.

Find the \(LD\widetilde{U}\)-factorization of the singular matrix \(\A\) by Algorithm 1 and then by Algorithm 2. What do you notice about the diagonals of \(\L\text{?}\) Of \(\U\text{?}\)

5.

\(\A=\left(\begin{array}{rrr}3\amp -1\amp 4\\-3\amp 1\amp -4\\6\amp -2\amp 8 \end{array} \right)\)

6.

\(\A=\left(\begin{array}{rrr}1\amp -2\amp -4\\2\amp -4\amp -8\\-3\amp 6\amp 12 \end{array} \right)\)

Exercise Group.

Find the \(LDU_1\)-factorization of \(\A\text{.}\)

7.

\(\A=\left(\begin{array}{rrr}3\amp 0\amp 0\\9\amp -6\amp 0\\-12\amp 6\amp 7 \end{array} \right)\)

8.

\(\A=\left(\begin{array}{rrr}1\amp 0\amp 0\\2\amp -4\amp 0\\-3\amp 6\amp 12 \end{array} \right)\)

9.

\(\A=\left(\begin{array}{rrr}-5\amp 0\amp 7\\0\amp -3\amp 1\\0\amp 0\amp 4 \end{array} \right)\)

10.

\(\A=\left(\begin{array}{rrr}-2\amp 2\amp 3\\0\amp -2\amp 1\\0\amp 0\amp 7 \end{array} \right)\)

11.

Prove Theorem 3.13.7.

12.

Prove Theorem 3.13.9.

13.

Consider \(\A=\left(\begin{array}{rrr}0\amp -2\amp -4\\2\amp -3\amp -9\\-5\amp 16\amp 15 \end{array} \right)\text{.}\) Multiply \(\A\) on the left by a permutation matrix \(\P_{ijk}\) (see Definition 3.7.11) to resolve the problem of having a \(0\) in the \(1,1\)-position, then find the \(LU\)-factorization of the resulting matrix by either algorithm. Then find \(\P_{i'j'k'}\) for which \(\A=\P_{i'j'k'}\L\U\text{.}\)