# Solving Eigenvector Order Homework

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In summary, an eigenvector is a vector that, when multiplied by a matrix, results in a scalar multiple of itself. It is significant in understanding the behavior of a system and predicting its future state. To find eigenvectors, we need to find the eigenvalues first and then solve for the corresponding eigenvectors using different methods. Eigenvectors can be complex numbers and are used in data analysis to reduce the dimensionality of a dataset.
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## Homework Statement

A =
2 -2
-2 5

Eigenvalues are: 6, 1

Find eigenvectors. My only question is about order. My book lists them in the opposite order as I and I am not sure where I went wrong.

## Homework Equations

A =
2 -2
-2 5

Eigenvalues are: 6, 1

## The Attempt at a Solution

i..e for 1:
1λ1 -2λ2 = 0
λ1 = 2λ2
= (2λ2)/(λ2)

Let λ2 = 1

= 2/√5, 1/√5

Yet my book lists them as 1/√5, 2/√5. Why is this?

## 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 vector remains the same, but its magnitude changes.

## 2. What is the significance of solving eigenvector order homework?

Solving eigenvector order homework helps us in understanding the behavior of a system and predicting its future state. It is also used in various fields such as physics, engineering, and computer science to analyze data and make informed decisions.

## 3. How do you find eigenvectors?

To find eigenvectors, we first need to find the eigenvalues by solving the characteristic equation of the matrix. Once we have the eigenvalues, we can plug them back into the original matrix and solve for the corresponding eigenvectors using Gaussian elimination or other methods.

## 4. Can eigenvectors be complex numbers?

Yes, eigenvectors can be complex numbers. In fact, when a matrix has complex eigenvalues, the corresponding eigenvectors will also be complex numbers.

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

Eigenvectors are used in data analysis to reduce the dimensionality of a dataset. By finding the eigenvectors of a covariance matrix, we can identify the most important features or variables in the data and use them to represent the dataset in a lower-dimensional space.

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