Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question
![Suppose \( Q \) is the quadratic form below:
\[
Q = \mathbf{x}^T A \mathbf{x}, \quad A = \begin{pmatrix} 5 & 2 & 1 \\ 2 & 8 & 2 \\ 1 & 2 & 5 \end{pmatrix}
\]
The minimum value of \( Q \) subject to \( \mathbf{x}^T \mathbf{x} = 1 \) is \( Q = 4 \). An eigenvector of \( A \) associated with eigenvalue \( \lambda = 4 \) is
\[
\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ c \end{pmatrix}
\]
What is \( c \) equal to? \[ \boxed{-2} \]
An eigenvector associated with eigenvalue \( \lambda = 4 \) is \( \mathbf{v} = (0, 1, -2)^T \). The minimum value of \( Q \), subject to \( \mathbf{x}^T \mathbf{x} = 1 \), is obtained at \( \mathbf{x}_0 \), where:
\[
\mathbf{x}_0 = \begin{pmatrix} 0 \\ k_0 \\ k_1 \end{pmatrix}
\]
If \( \mathbf{v} \) is parallel to \( \mathbf{x}_0 \) and \( k_0 > 0 \), what must \( k_1 \) be equal to? (answer must contain at least 3 decimal places)
\[ \boxed{-2} \]](https://content.bartleby.com/qna-images/question/8b609ed5-0879-4e23-8eee-4515e84ef2b9/e73538ee-0266-4617-bedb-c1d817e31345/i46gifj_processed.png)
Transcribed Image Text:Suppose \( Q \) is the quadratic form below:
\[
Q = \mathbf{x}^T A \mathbf{x}, \quad A = \begin{pmatrix} 5 & 2 & 1 \\ 2 & 8 & 2 \\ 1 & 2 & 5 \end{pmatrix}
\]
The minimum value of \( Q \) subject to \( \mathbf{x}^T \mathbf{x} = 1 \) is \( Q = 4 \). An eigenvector of \( A \) associated with eigenvalue \( \lambda = 4 \) is
\[
\mathbf{v} = \begin{pmatrix} 0 \\ 1 \\ c \end{pmatrix}
\]
What is \( c \) equal to? \[ \boxed{-2} \]
An eigenvector associated with eigenvalue \( \lambda = 4 \) is \( \mathbf{v} = (0, 1, -2)^T \). The minimum value of \( Q \), subject to \( \mathbf{x}^T \mathbf{x} = 1 \), is obtained at \( \mathbf{x}_0 \), where:
\[
\mathbf{x}_0 = \begin{pmatrix} 0 \\ k_0 \\ k_1 \end{pmatrix}
\]
If \( \mathbf{v} \) is parallel to \( \mathbf{x}_0 \) and \( k_0 > 0 \), what must \( k_1 \) be equal to? (answer must contain at least 3 decimal places)
\[ \boxed{-2} \]
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I should have noted that I also tried -2.000, it is also incorrect.
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The image displays that K1 = -2 is wrong. I got the same answer in my own work and still confused on what I'm doing wrong.
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Follow-up Question
I should have noted that I also tried -2.000, it is also incorrect.
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by Bartleby Expert
Follow-up Question
The image displays that K1 = -2 is wrong. I got the same answer in my own work and still confused on what I'm doing wrong.
Solution
by Bartleby Expert
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