# Linear Algebra Help: Calculating Eigenvalues & Eigenvectors of Matrix A

• GregoryGr
In summary, the conversation discusses calculating the eigenvalues and eigenvectors of a given matrix. The individual is looking for a way to make the determinant triangular, but it is not necessary for finding the values. The suggested method is to reorder the rows and then perform row-reduction to upper-triangular form. A resource for finding eigenvectors is also mentioned.
GregoryGr

## Homework Statement

Calculate the eigenvalues and eigenvectors of the matrix:
$$A= \begin{bmatrix} 3 & 2 & 2 &-4 \\ 2 & 3 & 2 &-1 \\ 1 & 1 & 2 &-1 \\ 2 & 2 & 2 &-1 \end{bmatrix}$$

nothing

## The Attempt at a Solution

I've found the eigenvalues, but what disturbes me, is that I can't find a way to make the determinant triangular, as to find the values faster. Can anybody see a way to do that?

You wouldn't make the determinant triangular, the determinant is just one number.

You can make the matrix triangular by row-reduction:
- number the rows top to bottom 1-4.
- reorder the rows: 3-2-4-1 --> 1-2-3-4
- after that the row-reduction to upper-triangular form should come easily.

You probably want to do this for each eigenvalue to find the eigenvectors - so the best order for the rows will be different each time.

You want to try this for the eigenvectors - consider:
http://www.millersville.edu/~bikenaga/linear-algebra/eigenvalue/eigenvalue.html

Last edited:

## What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with the study of vectors, matrices, and linear transformations. It is used to solve systems of linear equations and to analyze geometric shapes and data sets.

## What are Eigenvalues and Eigenvectors?

Eigenvalues and Eigenvectors are concepts in Linear Algebra that are used to understand how a linear transformation affects a vector. Eigenvalues are scalar values that represent the amount by which a vector is stretched or compressed by a transformation, while Eigenvectors are the corresponding vectors that are only scaled by the transformation.

## How do I calculate Eigenvalues and Eigenvectors of a matrix?

To calculate the Eigenvalues and Eigenvectors of a matrix, you can use the characteristic equation or the characteristic polynomial. First, find the characteristic polynomial by subtracting the identity matrix from the original matrix and taking the determinant. Then, solve for the Eigenvalues by setting the characteristic polynomial equal to zero. Finally, plug in the Eigenvalues into the original matrix to find the corresponding Eigenvectors.

## Why are Eigenvalues and Eigenvectors important?

Eigenvalues and Eigenvectors are important because they allow us to analyze the behavior of linear transformations and to find special directions in which the transformation has a simple effect. They are also used in various applications such as data compression, image processing, and quantum mechanics.

## What are some practical applications of calculating Eigenvalues and Eigenvectors?

Calculating Eigenvalues and Eigenvectors has many practical applications in fields such as physics, engineering, and computer science. Some examples include analyzing patterns in large data sets, reducing the dimensionality of data, and solving differential equations in physics and engineering problems.

Replies
2
Views
937
Replies
5
Views
899
Replies
1
Views
653
Replies
19
Views
3K
Replies
19
Views
699
Replies
4
Views
732
Replies
8
Views
2K
Replies
1
Views
1K
Replies
2
Views
787
Replies
7
Views
2K