Math Problem Statement
1 3 3 1 4 3 -1 3 4 Find eigen values and vectors
Solution
To find the eigenvalues and eigenvectors of the matrix , follow these steps:
Given matrix:
Step 1: Find the Eigenvalues
The eigenvalues are found by solving the characteristic equation:
Where is the identity matrix and is the eigenvalue.
- Subtract from the diagonal elements of :
- Calculate the determinant:
Expanding this determinant:
Simplify and solve for .
Step 2: Find the Eigenvectors
For each eigenvalue , solve:
Where is the eigenvector associated with .
Calculation:
Let me calculate the eigenvalues and eigenvectors.### Eigenvalues: The eigenvalues of the matrix are approximately:
Here, and are complex conjugates with very small imaginary parts, so they are almost real.
Eigenvectors:
The corresponding eigenvectors are:
For :
For :
For :
These eigenvectors are normalized (each has a magnitude of 1).
Would you like further details on the steps, or do you have any questions?
Related Questions:
- What is the significance of eigenvalues in a matrix?
- How are eigenvectors used in applications such as Principal Component Analysis (PCA)?
- Can you explain how the characteristic polynomial is derived from a matrix?
- What are some real-world applications of eigenvalues and eigenvectors?
- How do complex eigenvalues affect the stability of a system?
- What is the geometric interpretation of eigenvectors and eigenvalues?
- How can you verify if a vector is an eigenvector of a matrix?
- What happens if a matrix has repeated eigenvalues?
Tip: Always ensure eigenvectors are normalized when using them in applications, as their direction is often more important than their magnitude.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Eigenvalues
Eigenvectors
Characteristic Equation
Determinant
Formulas
Characteristic equation
Eigenvalue calculation
Eigenvector calculation
Theorems
Spectral theorem
Suitable Grade Level
College Level
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