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)\[ \boxed{-2} \]

Answered: Image /qna-images/answer/e73538ee-0266-4617-bedb-c1d817e31345.jpg

Suppose Q is the quadratic form below The minimum value of Q subject to = 1 is Q = 4. An eigenvector of A associated with eigenvalue λ = 4 is What is c equal to? -2 An eigenvector associated with eigenvalue λ = 4 is = (0, 1, -2). The minimum value of Q, subject to #= 1 is obtained at Zo, where: If is parallel to and k > 0, what must k₁ be equal to? (answer must contain at least 3 decimal places) -2 /5 2 1 Q = ¹ AZ, A = 2 8 2 1 2 5, -- (1) = to =

Suppose Q is the quadratic form below The minimum value of Q subject to = 1 is Q = 4. An eigenvector of A associated with eigenvalue λ = 4 is What is c equal to? -2 An eigenvector associated with eigenvalue λ = 4 is = (0, 1, -2). The minimum value of Q, subject to #= 1 is obtained at Zo, where: If is parallel to and k > 0, what must k₁ be equal to? (answer must contain at least 3 decimal places) -2 /5 2 1 Q = ¹ AZ, A = 2 8 2 1 2 5, -- (1) = to =

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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)

\[ \boxed{-2} \]

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