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Chapter 6: Q-6-15SE (page 331)

Exercises 15 and 16 concern the (real) Schur factorization of an \(n \times n\) matrix A in the form \(A = UR{U^T}\), where U is an orthogonal matrix and R is an \(n \times n\) upper triangular matrix.

15. Show that if A admits a (real) Schur factorization, \(A = UR{U^T}\), then A has \(n\) real eigenvalues, counting multiplicities.

Short Answer

Expert verified

It is proved that A has \(n\) real eigenvalues, counting multiplicities.

Step by step solution

01

Statement of Theorem 4 in Section 5.2

Theorem 4states that when \(n \times n\) matrices \(A\) and \(B\) are identical then the characteristic polynomial andeigenvalues of A and B are the same (with the same multiplicities).

02

Show that A has \(n\) real eigenvalues, counting multiplicities

Consider that \(A = UR{U^T}\). According to the hypothesis, \(U\) is orthogonal, therefore \({U^T} = {U^{ - 1}}\) and

\(\begin{array}{c}A - \lambda I = UR{U^T} - \lambda I\\ = UR{U^T} - \lambda UI{U^T}\\ = U\left( {R - \lambda I} \right){U^T}\end{array}\)

Hence,

\(\begin{array}{c}\det \left( {A - \lambda I} \right) = \det \left( {U\left( {R - \lambda I} \right){U^T}} \right)\\ = \det \left( U \right)\det \left( {R - \lambda I} \right)\det \left( {{U^T}} \right)\\ = \det \left( {R - \lambda I} \right)\end{array}\)

The characteristic polynomial for \(A\) and \(R\) is the same.

According to the hypothesis, \(R\) is upper triangular, therefore, the eigenvalues of R are its \(n\) diagonal entries. Therefore, \(A\) contains the same eigenvalues as R, according to theorem 4 in section 5.2. Hence, A has \(n\) real eigenvalues, counting multiplicities.

Thus, it is proved that A has \(n\) real eigenvalues, counting multiplicities.

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Most popular questions from this chapter

A certain experiment produces the data \(\left( {1,1.8} \right),\left( {2,2.7} \right),\left( {3,3.4} \right),\left( {4,3.8} \right),\left( {5,3.9} \right)\). Describe the model that produces a least-squares fit of these points by a function of the form

\(y = {\beta _1}x + {\beta _2}{x^2}\)

Such a function might arise, for example, as the revenue from the sale of \(x\) units of a product, when the amount offered for sale affects the price to be set for the product.

a. Give the design matrix, the observation vector, and the unknown parameter vector.

b. Find the associated least-squares curve for the data.

Compute the quantities in Exercises 1-8 using the vectors

\({\mathop{\rm u}\nolimits} = \left( {\begin{aligned}{*{20}{c}}{ - 1}\\2\end{aligned}} \right),{\rm{ }}{\mathop{\rm v}\nolimits} = \left( {\begin{aligned}{*{20}{c}}4\\6\end{aligned}} \right),{\rm{ }}{\mathop{\rm w}\nolimits} = \left( {\begin{aligned}{*{20}{c}}3\\{ - 1}\\{ - 5}\end{aligned}} \right),{\rm{ }}{\mathop{\rm x}\nolimits} = \left( {\begin{aligned}{*{20}{c}}6\\{ - 2}\\3\end{aligned}} \right)\)

5. \(\left( {\frac{{{\mathop{\rm u}\nolimits} \cdot {\mathop{\rm v}\nolimits} }}{{{\mathop{\rm v}\nolimits} \cdot {\mathop{\rm v}\nolimits} }}} \right){\mathop{\rm v}\nolimits} \)

In Exercises 13 and 14, the columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that \(A = QR\). Check your work.

13. \(A = \left( {\begin{aligned}{{}{}}5&9\\1&7\\{ - 3}&{ - 5}\\1&5\end{aligned}} \right),{\rm{ }}Q = \left( {\begin{aligned}{{}{}}{\frac{5}{6}}&{ - \frac{1}{6}}\\{\frac{1}{6}}&{\frac{5}{6}}\\{ - \frac{3}{6}}&{\frac{1}{6}}\\{\frac{1}{6}}&{\frac{3}{6}}\end{aligned}} \right)\)

In Exercises 7鈥�10, let\[W\]be the subspace spanned by the\[{\bf{u}}\]鈥檚, and write y as the sum of a vector in\[W\]and a vector orthogonal to\[W\].

8.\[y = \left[ {\begin{aligned}{ - 1}\\4\\3\end{aligned}} \right]\],\[{{\bf{u}}_1} = \left[ {\begin{aligned}1\\1\\{\bf{1}}\end{aligned}} \right]\],\[{{\bf{u}}_2} = \left[ {\begin{aligned}{ - 1}\\3\\{ - 2}\end{aligned}} \right]\]

A simple curve that often makes a good model for the variable costs of a company, a function of the sales level \(x\), has the form \(y = {\beta _1}x + {\beta _2}{x^2} + {\beta _3}{x^3}\). There is no constant term because fixed costs are not included.

a. Give the design matrix and the parameter vector for the linear model that leads to a least-squares fit of the equation above, with data \(\left( {{x_1},{y_1}} \right), \ldots ,\left( {{x_n},{y_n}} \right)\).

b. Find the least-squares curve of the form above to fit the data \(\left( {4,1.58} \right),\left( {6,2.08} \right),\left( {8,2.5} \right),\left( {10,2.8} \right),\left( {12,3.1} \right),\left( {14,3.4} \right),\left( {16,3.8} \right)\) and \(\left( {18,4.32} \right)\), with values in thousands. If possible, produce a graph that shows the data points and the graph of the cubic approximation.
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