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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q11E (page 413)

The singular values of any triangular matrix are the absolute values of its diagonal entries.

Short Answer

Expert verified

The given statement is FALSE.

Step by step solution

01

Step 1: Definition of singular values:

The diagonal entries of the matrix are singular values, which are arranged in ascending order.

The singular values are found from the non- zero eigenvalues of $A^{T} A$ .

Hence, singular value is $蟽 = \sqrt{位}$ .

02

Step 2: To Find TRUE or FALSE

Let

$A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$

Then, $A$ is upper triangular.

Let's check if both the singular values of $A$ are l or not.

$A^{T} A = [\begin{matrix} 1 & 0 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] A^{T} A = [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}]$

Find the singular values:

$(A^{T} A - 位 I) = 0$

$[\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}] - 位 [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] = 0 [\begin{matrix} 1 & 1 \\ 1 & 2 \end{matrix}] - [\begin{matrix} 位 & 0 \\ 0 & 位 \end{matrix}] = 0 [\begin{matrix} 1 - 位 & 1 \\ 1 & 2 - 位 \end{matrix}] = 0 (1 - 位) (2 - 位) - (1) (1) = 0$

The characteristic polynomial is

$A^{T} A = (1 - 位) (2 - 位) - 1 = 位^{2} - 3 位 + 1$

The roots of the polynomial are $\frac{3 卤 \sqrt{5}}{2}$ .

The singular values are found from the non- zero eigenvalues of $A^{T} A$ .

Hence, singular value $蟽 = \sqrt{位}$ .Thus, the singular values of $A$ cannot be equal to 1.

Thus, the given statement is FALSE.

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Most popular questions from this chapter

If A is an $n 脳 n$ matrix, what is the product of its singular values $蟽_{1}, . . ., 蟽_{n}$ ? State the product in terms of the determinant of A. For a $2 脳 2$ matrix A, explain this result in terms of the image of the unit circle.

All positive definite matrices are invertible.

Find the singular values of $A = [\begin{matrix} 1 & 2 \\ 2 & 4 \end{matrix}]$ . Find a unit vector $\overset{鈫�}{v}$ such that $|| A \overset{鈫�}{v_{1}} || = 蟽_{1}$ . Sketch the image of the unit circle.

37. If Ais a positive definite $n 脳 n$ matrix and $\overset{鈫�}{x}$ is a nonzero vector in $R^{n}$ , then the angle between $\overset{鈫�}{x}$ and $A \overset{鈫�}{x}$ must be acute.

Sketch the curves defined in Exercises 15 through 20. In each case, draw and label the principal axes, label the intercepts of the curve with the principal axes, and give the formula of the curve in the coordinate system defined by the principal axes.

18. $9 x_{1}^{2} 路 4 x_{1} x_{2} + 6 x_{2}^{2} = 1$

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