4 51 Find the eigenvalues and eigenvectors of the matrix A = -10 d1 =, v1 = and
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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![**Finding Eigenvalues and Eigenvectors of the Matrix**
Consider the matrix \( A = \begin{bmatrix} 4 & 5 \\ -10 & -6 \end{bmatrix} \).
To find the eigenvalues (\( \lambda \)) and eigenvectors (\( \vec{v} \)) of this matrix, perform the following steps:
1. **Find the Eigenvalues \(\lambda_1\) and \(\lambda_2\)**:
- Determine the characteristic equation by setting \(\det(A - \lambda I) = 0\).
- Solve for \(\lambda\) to find \(\lambda_1\) and \(\lambda_2\).
2. **Find the Eigenvectors \(\vec{v}_1\) and \(\vec{v}_2\)**:
- For each eigenvalue \(\lambda\), substitute it back into the equation \((A - \lambda I)\vec{v} = \vec{0}\).
- Solve for the eigenvectors \(\vec{v}_1\) and \(\vec{v}_2\), ensuring that each vector is non-trivial.
Next, fill in the blanks:
- \(\lambda_1 = \) [ ]
- \(\vec{v}_1 = \begin{bmatrix} [ \ ] \\ [ \ ] \end{bmatrix}\)
and
- \(\lambda_2 = \) [ ]
- \(\vec{v}_2 = \begin{bmatrix} [ \ ] \\ [ \ ] \end{bmatrix}\)
**Diagrams and Calculations:**
- **Matrix \( A \):** Represents the linear transformation described by the matrix.
- The solution entails calculating the determinant for finding \(\lambda\), and solving linear equations for \(\vec{v}\).
Understanding eigenvalues and eigenvectors reveals information about the matrix's transformations, such as scaling and rotation in vector spaces.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70a05657-0307-4b7c-962c-ab3824846297%2F76b4f1be-2c98-440a-89ed-a257a6dec8ff%2Fggtuwo7_processed.png&w=3840&q=75)
Transcribed Image Text:**Finding Eigenvalues and Eigenvectors of the Matrix**
Consider the matrix \( A = \begin{bmatrix} 4 & 5 \\ -10 & -6 \end{bmatrix} \).
To find the eigenvalues (\( \lambda \)) and eigenvectors (\( \vec{v} \)) of this matrix, perform the following steps:
1. **Find the Eigenvalues \(\lambda_1\) and \(\lambda_2\)**:
- Determine the characteristic equation by setting \(\det(A - \lambda I) = 0\).
- Solve for \(\lambda\) to find \(\lambda_1\) and \(\lambda_2\).
2. **Find the Eigenvectors \(\vec{v}_1\) and \(\vec{v}_2\)**:
- For each eigenvalue \(\lambda\), substitute it back into the equation \((A - \lambda I)\vec{v} = \vec{0}\).
- Solve for the eigenvectors \(\vec{v}_1\) and \(\vec{v}_2\), ensuring that each vector is non-trivial.
Next, fill in the blanks:
- \(\lambda_1 = \) [ ]
- \(\vec{v}_1 = \begin{bmatrix} [ \ ] \\ [ \ ] \end{bmatrix}\)
and
- \(\lambda_2 = \) [ ]
- \(\vec{v}_2 = \begin{bmatrix} [ \ ] \\ [ \ ] \end{bmatrix}\)
**Diagrams and Calculations:**
- **Matrix \( A \):** Represents the linear transformation described by the matrix.
- The solution entails calculating the determinant for finding \(\lambda\), and solving linear equations for \(\vec{v}\).
Understanding eigenvalues and eigenvectors reveals information about the matrix's transformations, such as scaling and rotation in vector spaces.
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