Find Eigenvectors of 3x3 Matrix

In summary, to find the eigenvalues and an orthonormal set of eigenvectors for the given matrix, we need to find the determinant of A - xI and set it equal to 0. Simplifying, we get the equation (2-x)(x^2 - 2x - 1) = 0. This can be solved for the eigenvalues, which are 2, 1+i, and 1-i. To find the corresponding eigenvectors, we can substitute each eigenvalue into (A - xI)v = 0 and solve for v.
  • #1
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Homework Statement


Find the eigenvalues and an orthonormal set of eigenvectors for this matrix:

|1 1+i 0|
|1-i 1 0|
|0 0 2|

Homework Equations


Find the determinant of A - xI, where A is the matrix, I is the identity matrix, and x denotes eigenvalues
Set the determinant equal to 0, and then find eigenvectors for each eigenvalue


The Attempt at a Solution



If x denotes an eigenvalue, I found the determinant of this matrix to be (2-x)((1-x)(1-x) - (1+i)(1-i)) = (2-x)(x^2 - 2x - 1). Then I got stuck, since I do not know how to factor this equation. How do I find the eigenvalues for this matrix? Am I doing something wrong?
 
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  • #2
You've already factored it as much as you need to. What's left is a quadratic equation. You can solve that without factoring, can't you?
 
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1. What are eigenvectors and why are they important?

Eigenvectors are special vectors that, when multiplied by a matrix, give a scalar multiple of itself as the result. They are important because they represent the directions in which a linear transformation has a simple effect, making them useful for understanding and analyzing complex systems.

2. How do you find the eigenvectors of a 3x3 matrix?

To find the eigenvectors of a 3x3 matrix, you first need to find the eigenvalues by solving the characteristic equation. Then, for each eigenvalue, you can use the eigenvector formula to find the corresponding eigenvector. This involves setting up a system of linear equations and solving for the variables.

3. What is the purpose of finding eigenvectors?

The purpose of finding eigenvectors is to understand the behavior and properties of a linear transformation represented by a matrix. They can be used to simplify complex systems, identify patterns and symmetries, and make predictions about future outcomes.

4. Are there any real-world applications for finding eigenvectors?

Yes, there are many real-world applications for finding eigenvectors, such as in physics, engineering, and economics. For example, in physics, eigenvectors can be used to study the behavior of quantum mechanical systems. In economics, they can be used to analyze market trends and predict future economic conditions.

5. Can every matrix have eigenvectors?

No, not every matrix has eigenvectors. The matrix must be square (same number of rows and columns) and have distinct eigenvalues in order to have eigenvectors. If the matrix does not meet these requirements, it will not have eigenvectors.

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