# What should I do when faced with a mixture of real and complex eigenvalues?

• Mindscrape
In summary, the conversation discusses solving a differential equation and finding the eigenvalues by taking the determinant and using a "basketweave" method. The final solution involves a mixture of real and complex eigenvalues, with the real eigenvalue not having an eigenvector. The conversation ends with a suggestion to look at the matrix and pick the eigenvector for the real eigenvalue.
Mindscrape
I need to solve the differential equation
$$\mathbf{x'} = \left( \begin{array}{ccc} 3 & 0 & -1\\ 0 & -3 & -1\\ 0 & 2 & -1 \end{array} \right) \mathbf{x}$$

solving for the eigenvalues by taking the determinate and using the "basketweave" yields

$$(3 - \lambda)(-3-\lambda)(-1-\lambda) + 2(3-\lambda) = 0$$

and further simplification shows that

$$-\lambda^3 - \lambda^2 +7\lambda +15 = 0$$

guessing roots, I found that 3 is one and divided the polynomial by the root

$$(\lambda - 3)(-\lambda^2 - 4\lambda -5)=0$$

so the eigenvalues are (solving for both eqns in the brackets)
$$\lambda = 3$$
$$\lambda = 2 + i$$
$$\lambda = 2 - i$$

The thing is that when I put in the real eigenvalue, it doesn't seem to have an eigenvector. Is this right? I think it would be because I don't have any idea how to get the solution for a mixture of real and complex eigenvalues. The solutions I know how to solve are when the eigenvectors are of the form v1,v2=p±iq. Though I imagine that if the real eigenvalue made an eigenvector I could just take the general solution of that and add it to the general solution of the complex eigenvectors.

Basically what I want to know is, what do I do know?

there's some miscalculation there.
eigenvalues are 3, (-2 + i), (-2 -i)

you can just look at the matrix and
pick the eigenvector that gives 3 as
the eigenvalue. (1, 0, 0).

## 1. What is an eigenvector?

An eigenvector is a vector that does not change direction when multiplied by a given square matrix. It only changes its magnitude (or length) by a scalar factor, known as the eigenvalue.

## 2. What is an eigenvector problem?

An eigenvector problem is a mathematical problem that involves finding the eigenvectors and eigenvalues of a given square matrix. It is an important concept in linear algebra and is used in various fields of science, including physics and engineering.

## 3. Why is finding eigenvectors and eigenvalues important?

Finding eigenvectors and eigenvalues can provide insights into the behavior of a system represented by a matrix. They can help identify special directions or patterns in the data and can be used to simplify complex equations and calculations.

## 4. How do you solve an eigenvector problem?

To solve an eigenvector problem, you need to first find the eigenvalues by solving the characteristic equation. Then, for each eigenvalue, you need to find the corresponding eigenvector by solving a system of linear equations. This can be done manually or by using software such as MATLAB or Python.

## 5. What are some real-world applications of eigenvector problems?

Eigenvector problems are used in a wide range of applications, such as image and signal processing, data compression, and machine learning. They are also used in physics to study quantum mechanics and in engineering to analyze structural vibrations and electrical circuits.

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