Finding Eigenvectors with Close Eigenvalues

In summary, the conversation discusses how to find the eigenvalues and eigenvectors of a given matrix. The eigenvalues are -2.659, 0.329-.802i, and 0.329+.802i. The method to find the eigenvectors involves plugging each eigenvalue into the matrix and solving the system (A-xI)v=0. However, in some cases, this method may not yield accurate results and alternative methods may need to be used.
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Homework Statement



Given the characteristic polynomial -2+x-2x^2-x^3, find the eigenvalues and eigenvectors of the matrix [-1, -1, 0] [1, 1, 1] [3, 1, -2]

Homework Equations


The Attempt at a Solution



The eigenvalues are -2.659, 0.329-.802i, and 0.329+.802i. Next you plug each eigenvalue into the matrix A-xI to solve the system (A-xI)v=0 and find the eigenvectors. Then you solve the system by reducing the matrix to row echelon form. However, when I do that I get the identity matrix. So then what are the eigenvectors?
 
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  • #2
That occurs often when the eigenvalues are not close enough. You either need to switch to a less sensitive method, or find closer approximations to the eigenvalues.
 

What is an eigenvector?

An eigenvector is a vector that, when multiplied by a given square matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector remains unchanged after the matrix multiplication, and the scalar multiple is known as the eigenvalue.

Why are eigenvectors important?

Eigenvectors are important because they provide a way to understand how a matrix transformation affects different directions in space. They also play a crucial role in many fields of science and engineering, such as in physics, chemistry, and computer science.

How do you find eigenvectors?

To find eigenvectors, you first need to find the eigenvalues of the matrix by solving for the roots of the characteristic polynomial. Then, for each eigenvalue, you can find the corresponding eigenvector by solving a system of equations using the eigenvalue and the matrix. Alternatively, you can use computational methods, such as the power method or the QR algorithm, to find the eigenvectors.

Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors for the same eigenvalue. In fact, the number of linearly independent eigenvectors for a given eigenvalue is known as the geometric multiplicity of the eigenvalue.

What is the relationship between eigenvectors and eigenvalues?

The eigenvalues and eigenvectors of a matrix are closely related. Each eigenvalue corresponds to a unique eigenvector, and vice versa. Furthermore, the eigenvalues and eigenvectors can provide important information about the characteristics and behavior of the matrix transformation.

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