# Help with eigenvalues from matrix

• Sparky_
In summary, the conversation is about finding the eigenvalues for a given matrix A. The person asking the question is having trouble with the calculations, but eventually realizes their mistake and thanks the person helping them. The final answer is that the eigenvalues of A are -3, -4, and 5.
Sparky_

## Homework Statement

find the eignevalues (a part of a larger problem) for

A= | -4 1 1 |
| 1 5 -1 |
| 0 1 -3 |

## The Attempt at a Solution

= | -4-x 1 1 |
| 1 5-x -1 |
| 0 1 -3-x |

(-4-x)[ (5-x)(-3-x) + 1 ] - (1)[ (1)(-3-x) - 0] + (1)[1-0]

(-4-x)[ x^2 -2x -14] + x + 4

(-4)[-x^3 + 2x^2 + 14x] + x + 4

4x^3 - 8x^2 - 56x + x + 4

4x^3 - 8x^2 - 55x + 4

I do not see the roots for the above being -3, -4, and 5.

The book has x1 = -3, x2 = -4, and x3 = 5.

When I run the eigenvalue routine in the software package Maxima, I believe I am getting (-3 -4,5) - output is a little confusing. - anyway I am seeing the same numbers

When I expand (x+3)(x+4)(x-5)

I get 3x^3 - 3x^2 - 60x.

Help?

Thanks
Sparky_

Sparky_ said:
(-4-x)[ (5-x)(-3-x) + 1 ] - (1)[ (1)(-3-x) - 0] + (1)[1-0]

(-4-x)[ x^2 -2x -14] + x + 4

(-4)[-x^3 + 2x^2 + 14x] + x + 4

Found it. You're not properly multiplying (-4-x) by [ x^2 -2x -14]. Unfortunately, you can't just multiply by the -x and then by -4 (I wish you could). Instead:

(-4-x)[ x^2 -2x -14] = (-4)[ x^2 -2x -14] - x[ x^2 -2x -14] = ...

Crap - (embarrassed)

it's been too long since I've done this - out of practice

Thanks for the help!

## 1. What are eigenvalues and why are they important?

Eigenvalues are a concept in linear algebra that represent the scalars which the matrix can be multiplied by without changing the direction of the corresponding eigenvector. They are important because they can provide insights into the behavior of a system and can be used to solve equations and make predictions.

## 2. How do I find the eigenvalues of a matrix?

To find the eigenvalues of a matrix, you need to first calculate the determinant of the matrix. Then, you need to solve the characteristic equation, which is obtained by setting the determinant equal to 0. The solutions to this equation are the eigenvalues of the matrix.

## 3. What is the relationship between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are closely related. Eigenvectors are the vectors that do not change direction when multiplied by the corresponding eigenvalue. In other words, eigenvalues and eigenvectors are a pair and cannot exist without each other.

## 4. Can a matrix have complex eigenvalues?

Yes, a matrix can have complex eigenvalues. This can happen when the matrix has complex numbers as its elements. In this case, the eigenvalues will also be complex numbers.

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

Eigenvalues are used in data analysis to reduce the dimensionality of a dataset. This is done by finding the eigenvectors and eigenvalues of the covariance matrix of the data, and then selecting the eigenvectors with the highest corresponding eigenvalues. These eigenvectors can then be used as new variables to represent the data, while still retaining most of the important information.

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