Findtheeigenvaluesandeigenvectorsof: A= 3 0 0 0 5 0 0 0 7

The given matrix \( A \) is:

\[
A = \begin{pmatrix}
3 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 7
\end{pmatrix}
\]

This is a diagonal matrix, where the diagonal entries are \( 3 \), \( 5 \), and \( 7 \).

### Eigenvalues:
For a diagonal matrix, the eigenvalues are simply the diagonal elements. Hence, the eigenvalues of matrix \( A \) are:
\[
\lambda_1 = 3, \quad \lambda_2 = 5, \quad \lambda_3 = 7
\]

### Eigenvectors:
For a diagonal matrix, the eigenvectors correspond to the standard basis vectors. The eigenvector associated with each eigenvalue is a vector that has a 1 in the position corresponding to the eigenvalue and 0 in all other positions.

Thus, the eigenvectors for matrix \( A \) are:

- For \( \lambda_1 = 3 \), the eigenvector is:
  \[
  \mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}
  \]
  
- For \( \lambda_2 = 5 \), the eigenvector is:
  \[
  \mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}
  \]
  
- For \( \lambda_3 = 7 \), the eigenvector is:
  \[
  \mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}
  \]

### Summary:
- Eigenvalues: \( 3, 5, 7 \)
- Eigenvectors: 
  \[
  \mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad 
  \mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad
  \mathbf{v}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}
  \]

Would you like more details or further explanations?

Here are 5 related questions:
1. What is the geometric interpretation of eigenvalues and eigenvectors?
2. How would the process change for a non-diagonal matrix?
3. How do you verify that a vector is an eigenvector?
4. What does it mean for a matrix to be diagonalizable?
5. How are eigenvalues used in solving systems of differential equations?

**Tip:** For diagonal matrices, the eigenvalues are always the diagonal elements, making the calculation straightforward.

Find the eigenvalues and eigenvectors of: A = 3 0 0 0 5 0 0 0 7

Learn how to find the eigenvalues and eigenvectors of the diagonal matrix A = [[3, 0, 0], [0, 5, 0], [0, 0, 7]]. This solution explains key linear algebra concepts, including the formula for eigenvalues and eigenvectors. Suitable for college-level students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation