0 -6 6 -2 0 0 -4 2 has two distinct real eigenvalues ₁ < ₂. Find the eigenvalues and a basis for each eigenspace. The smaller eigenvalue ₁ is The larger eigenvalue λ₂ is A = -2 0 0 0 t NNO 0 2 and a basis for its associated eigenspace is 100 and a basis for its associated eigenspace is
0 -6 6 -2 0 0 -4 2 has two distinct real eigenvalues ₁ < ₂. Find the eigenvalues and a basis for each eigenspace. The smaller eigenvalue ₁ is The larger eigenvalue λ₂ is A = -2 0 0 0 t NNO 0 2 and a basis for its associated eigenspace is 100 and a basis for its associated eigenspace is
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
![The matrix
\[ A = \begin{bmatrix} -2 & 0 & -6 & 6 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & -4 & 2 \\ 0 & 0 & -4 & 2 \end{bmatrix} \]
has two distinct real eigenvalues \(\lambda_1 < \lambda_2\). Find the eigenvalues and a basis for each eigenspace.
The smaller eigenvalue \(\lambda_1\) is \(\_\_\_\_\_\) and a basis for its associated eigenspace is
\[
\left\{
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix},
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix},
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix}
\right\}.
\]
The larger eigenvalue \(\lambda_2\) is \(\_\_\_\_\_\) and a basis for its associated eigenspace is
\[
\left\{
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix}
\right\}.
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b2b0246-74c5-427f-bbae-6a253749f017%2F0bee0c12-b385-412b-9abb-ad316dc90ac3%2Fbbqph6r_processed.png&w=3840&q=75)
Transcribed Image Text:The matrix
\[ A = \begin{bmatrix} -2 & 0 & -6 & 6 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & -4 & 2 \\ 0 & 0 & -4 & 2 \end{bmatrix} \]
has two distinct real eigenvalues \(\lambda_1 < \lambda_2\). Find the eigenvalues and a basis for each eigenspace.
The smaller eigenvalue \(\lambda_1\) is \(\_\_\_\_\_\) and a basis for its associated eigenspace is
\[
\left\{
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix},
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix},
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix}
\right\}.
\]
The larger eigenvalue \(\lambda_2\) is \(\_\_\_\_\_\) and a basis for its associated eigenspace is
\[
\left\{
\begin{bmatrix}
\_\_\_\\
\_\_\_\\
\_\_\_\\
\_\_\_
\end{bmatrix}
\right\}.
\]
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