# Find eigenvalues & eigenvectors

• rugerts
In summary, the conversation discusses the use of a trick to quickly find eigenvectors for simple 2x2 cases, but there is confusion about whether the eigenvectors shown in the blue box are equivalent to one another. The expert confirms that they are the same up to a constant and explains how to verify the correctness of the eigenvalues and eigenvectors.
rugerts
Homework Statement
Find eigenvalues & eigenvectors
Relevant Equations
det(A-r*I) = 0
Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at least for simple 2x2 cases like these) by simply taking a row (of the matrix after having plugged in the eigenvalue) and shaping it such that the leftmost value is the first value for the eigenvector (with opposite sign) and the second eigenvector value is the same as the rightmost value. I heard that this is valid for either row choice... but when I do this, the answers appear to be different (hence the question mark over the inequality). So, my question is, are these eigenvectors shown in the blue box equivalent to one another? Does this trick really work? It seems to.

Sort of. Those are the same two eigenvectors up to a constant, but in opposite order.

Multiply your ##\vec v## by the constant (1 + i). As you know, any multiple of an eigenvector is still an eigenvector of the same eigenvalue.
$$(1 + i) {2 \choose {2 - 2i}} = {{2 + 2i} \choose {2(1 - i)(1+i)}} = {{2+2i} \choose 4}$$
and similarly multiply ##\vec w## by the constant (1 - i).
$$(1 - i) {2 \choose {2 + 2i}} = {{2 - 2i} \choose {2(1 + i)(1-i)}} = {{2-2i} \choose 4}$$
Since the order is reversed, so are the eigenvalues with which each is associated.

rugerts
Also, you can verify that your eigenvalues and eigenvectors are correct or not by plugging them into the matrix equation and determining whether ##(A - \lambda I) \vec x = \vec 0##.

RPinPA said:
Sort of. Those are the same two eigenvectors up to a constant, but in opposite order.

Multiply your ##\vec v## by the constant (1 + i). As you know, any multiple of an eigenvector is still an eigenvector of the same eigenvalue.
$$(1 + i) {2 \choose {2 - 2i}} = {{2 + 2i} \choose {2(1 - i)(1+i)}} = {{2+2i} \choose 4}$$
and similarly multiply ##\vec w## by the constant (1 - i).
$$(1 - i) {2 \choose {2 + 2i}} = {{2 - 2i} \choose {2(1 + i)(1-i)}} = {{2-2i} \choose 4}$$
Since the order is reversed, so are the eigenvalues with which each is associated.
Ahh, I see. Thank you

## What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to analyze linear transformations. Eigenvalues are scalar values that represent the amount by which an eigenvector is stretched or compressed by a transformation. Eigenvectors are non-zero vectors that remain in the same direction after being transformed.

## Why is it important to find eigenvalues and eigenvectors?

Finding eigenvalues and eigenvectors can help us understand how a linear transformation affects a vector. They can also be used to simplify complex calculations and solve systems of linear equations.

## How do you find eigenvalues and eigenvectors?

To find eigenvalues and eigenvectors, we first need to create a matrix from the coefficients of the linear transformation. Then, we solve for the eigenvalues by finding the roots of the characteristic polynomial of the matrix. Finally, we use the eigenvalues to find the corresponding eigenvectors.

## What is the relationship between eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are closely related. Each eigenvalue has a corresponding eigenvector, and the eigenvector represents the direction in which the transformation is acting. The eigenvalue represents the amount by which the eigenvector is stretched or compressed by the transformation.

## How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors have many real-world applications, such as in physics, engineering, and computer graphics. They are used to analyze the behavior of systems, such as in quantum mechanics, and to find important features in data, such as in image processing. They can also be used to solve differential equations and model complex systems.

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