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Chapter 2: Q2.5-16Q (page 93)

Find an LU factorization of the matrices in Exercise 7-16 (with L unit lower triangular). Note that MATLAB will usually produce a permuted LU factorization because it uses partial pivoting for numerical accuracy.

\[\left[ {\begin{array}{*{20}{c}}{\bf{2}}&{ - {\bf{6}}}&{\bf{6}}\\{ - {\bf{4}}}&{\bf{5}}&{ - {\bf{7}}}\\{\bf{3}}&{\bf{5}}&{ - {\bf{1}}}\\{ - {\bf{6}}}&{\bf{4}}&{ - {\bf{8}}}\\{\bf{8}}&{ - {\bf{3}}}&{\bf{9}}\end{array}} \right]\]

Short Answer

Expert verified

\[\left[ {\begin{array}{*{20}{c}}1&0&0&0&0\\{ - 2}&1&0&0&0\\{\frac{3}{2}}&{ - 2}&1&0&0\\{ - 3}&2&0&1&0\\4&{ - 3}&0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&0&0\\0&0&0\\0&0&0\end{array}} \right]\]

Step by step solution

01

Apply the row operation in the given matrix

Let \[A = \left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\{ - 4}&5&{ - 7}\\3&5&{ - 1}\\{ - 6}&4&{ - 8}\\8&{ - 3}&9\end{array}} \right]\].

Apply the row operation to reduce the matrix into an upper triangular matrix.

At row 5, multiply row 1 by 4 and subtract it from row 5, i.e., \({R_5} \to {R_5} - 4{R_1}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\{ - 4}&5&{ - 7}\\3&5&{ - 1}\\{ - 6}&4&{ - 8}\\0&{21}&{ - 15}\end{array}} \right]\]

At row 4, multiply row 1 by 3 and add it to row 4, i.e., \({R_4} \to {R_4} + 3{R_1}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\{ - 4}&5&{ - 7}\\3&5&{ - 1}\\0&{ - 14}&{10}\\0&{21}&{ - 15}\end{array}} \right]\]

At row 3, multiply row 1 by \(\frac{3}{2}\) and subtract it from row 3, i.e., \({R_3} \to {R_3} - \frac{3}{2}{R_1}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\{ - 4}&5&{ - 7}\\0&{14}&{ - 10}\\0&{ - 14}&{10}\\0&{21}&{ - 15}\end{array}} \right]\]

At row 2, multiply row 1 by 2 and add it to row 2, i.e., \({R_2} \to {R_2} + 2{R_1}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&{14}&{ - 10}\\0&{ - 14}&{10}\\0&{21}&{ - 15}\end{array}} \right]\]

02

Apply the row operation in the given matrix

At row 5, multiply row 2 by 3 and add it to row 5, i.e., \({R_5} \to {R_5} + 3{R_2}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&{14}&{ - 10}\\0&{ - 14}&{10}\\0&0&0\end{array}} \right]\]

At row 4, add rows 3 and 4, i.e., \({R_4} \to {R_4} + {R_3}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&{14}&{ - 10}\\0&0&0\\0&0&0\end{array}} \right]\]

At row 3, multiply row 2 by 2 and add it to row 3, i.e., \({R_3} \to {R_3} + 2{R_2}\).

\[\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&0&0\\0&0&0\\0&0&0\end{array}} \right]\]

03

Calculate matrix L using the pivoted column of U

Using matrix \(A\), matrix Lcan be written as

\[\left[ {\begin{array}{*{20}{c}}2&0&0&0&0\\{ - 4}&{ - 7}&0&0&0\\3&{14}&1&0&0\\{ - 6}&{ - 14}&0&1&0\\8&{21}&0&0&1\end{array}} \right]\].

Divide column 1 by 2 and column 2 by \( - 7\).

\[\left[ {\begin{array}{*{20}{c}}1&0&0&0&0\\{ - 2}&1&0&0&0\\{\frac{3}{2}}&{ - 2}&1&0&0\\{ - 3}&2&0&1&0\\4&{ - 3}&0&0&1\end{array}} \right]\]

04

Write the product \(LU\)

The product of lower and upper triangular matrices can be written as

\[LU = \left[ {\begin{array}{*{20}{c}}1&0&0&0&0\\{ - 2}&1&0&0&0\\{\frac{3}{2}}&{ - 2}&1&0&0\\{ - 3}&2&0&1&0\\4&{ - 3}&0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&0&0\\0&0&0\\0&0&0\end{array}} \right]\].

So, the product \(LU\) is \[\left[ {\begin{array}{*{20}{c}}1&0&0&0&0\\{ - 2}&1&0&0&0\\{\frac{3}{2}}&{ - 2}&1&0&0\\{ - 3}&2&0&1&0\\4&{ - 3}&0&0&1\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2&{ - 6}&6\\0&{ - 7}&5\\0&0&0\\0&0&0\\0&0&0\end{array}} \right]\].

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Most popular questions from this chapter

If A, B, and X are \(n \times n\) invertible matrices, does the equation \({C^{ - 1}}\left( {A + X} \right){B^{ - 1}} = {I_n}\) have a solution, X? If so, find it.

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5鈥�8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

7. \[\left[ {\begin{array}{*{20}{c}}X&{\bf{0}}&{\bf{0}}\\Y&{\bf{0}}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&Z\\{\bf{0}}&{\bf{0}}\\B&I\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\{\bf{0}}&I\end{array}} \right]\]

In Exercises 27 and 28, view vectors in \({\mathbb{R}^n}\) as \(n \times 1\) matrices. For \({\mathop{\rm u}\nolimits} \) and \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^n}\), the matrix product \({{\mathop{\rm u}\nolimits} ^T}v\) is a \(1 \times 1\) matrix, called the scalar product, or inner product, of u and v. It is usually written as a single real number without brackets. The matrix product \({{\mathop{\rm uv}\nolimits} ^T}\) is an \(n \times n\) matrix, called the outer product of u and v. The products \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \) and \({{\mathop{\rm uv}\nolimits} ^T}\) will appear later in the text.

27. Let \({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 2}\\3\\{ - 4}\end{aligned}} \right)\) and \({\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}a\\b\\c\end{aligned}} \right)\). Compute \({{\mathop{\rm u}\nolimits} ^T}{\mathop{\rm v}\nolimits} \), \({{\mathop{\rm v}\nolimits} ^T}{\mathop{\rm u}\nolimits} \),\({{\mathop{\rm uv}\nolimits} ^T}\), and \({\mathop{\rm v}\nolimits} {{\mathop{\rm u}\nolimits} ^T}\).

In exercise 5 and 6, compute the product \(AB\) in two ways: (a) by the definition, where \(A{b_{\bf{1}}}\) and \(A{b_{\bf{2}}}\) are computed separately, and (b) by the row-column rule for computing \(AB\).

\(A = \left( {\begin{aligned}{*{20}{c}}{\bf{4}}&{ - {\bf{2}}}\\{ - {\bf{3}}}&{\bf{0}}\\{\bf{3}}&{\bf{5}}\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{\bf{3}}\\{\bf{2}}&{ - {\bf{1}}}\end{aligned}} \right)\)

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).
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