# Finding eigenvectors of a simple 2x2 matrix

• The thinker
In summary, the conversation is about solving for eigenvectors using the given matrix and eigenvalues. The participant is struggling with finding the correct solution and the expert suggests using a constant value for x to solve for y.
The thinker

The matrix is:
|1 2|
|3 4|

## The Attempt at a Solution

I've worked out the eigenvalues to be $$\stackrel{\underline{5\pm\sqrt{33}}}{2}$$

But when I plug the first eigenvalue back in I get:

|1 - $$\stackrel{\underline{5+\sqrt{33}}}{2}$$......2 |
|3......4 - $$\stackrel{\underline{5+\sqrt{33}}}{2}$$ |

Multiplied by (x,y) = (0,0)

[Sorry I couldn't fit that on the same line as the matrix.]Which in turn gives two equations:

x*(1-$$\stackrel{\underline{5+\sqrt{33}}}{2}$$) + 2y = 0

and3x + y*(4-$$\stackrel{\underline{5+\sqrt{33}}}{2}$$) = 0Which I can't work out how to solve for the eigenvector.

Is the only solution x=y=0?

You solve them like you solve any two linear equations. Remember if (x,y) is an eigenvector then a*(x,y) is also an eigenvector. So you are going to have an undetermined parameter which you may as well may make x. Put say, x=1 and solve for y. You should get the same value of y from both equations, since one equation is really a multiple of the other.

Thats exactly what I've been trying to do but X and/or Y always end up canceling and I just get some useless number.

The thinker said:

Thats exactly what I've been trying to do but X and/or Y always end up canceling and I just get some useless number.

Are you trying to eliminate a variable from the two equations and solve for the other? That won't work. If you eliminate x then the resulting equation is 0*y=0. Is that your problem? Like I said before, that's happening because the two equations are really the same. They have an infinite number of solutions. Put x equal to any constant (like x=1) and then solve for y.

AH!

Thank you!

## 1. What is an eigenvector?

An eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of itself. In other words, the direction of the eigenvector does not change when multiplied by the matrix, only the magnitude is scaled.

## 2. Why is finding eigenvectors important?

Eigenvectors are important in many areas of science and mathematics, including physics, engineering, and data analysis. They allow us to understand and describe the behavior of systems, such as the movement of objects or the spread of information.

## 3. How do you find eigenvectors of a 2x2 matrix?

To find the eigenvectors of a 2x2 matrix, we first need to find the eigenvalues by solving the characteristic equation. Then, we can plug these eigenvalues into the matrix and solve for the corresponding eigenvectors using Gaussian elimination or another method.

## 4. Can a 2x2 matrix have more than two eigenvectors?

No, a 2x2 matrix can have at most two eigenvectors. This is because the number of eigenvectors is equal to the number of distinct eigenvalues, and a 2x2 matrix can have at most two distinct eigenvalues.

## 5. How are eigenvectors used in data analysis?

In data analysis, eigenvectors are commonly used in a technique called Principal Component Analysis (PCA). This involves finding the eigenvectors of a covariance matrix to identify the most important features of a dataset. These eigenvectors can then be used to reduce the dimensionality of the data and simplify analysis.

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