- #1
rugerts
- 153
- 11
- Homework Statement
- Find eigenvalues & eigenvectors
- Relevant Equations
- det(A-r*I) = 0
Thanks for your time.
Find eigenvalues & eigenvectors
- Thread starter rugerts
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- Eigenvalues Eigenvectors
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.
Thanks for your time.
- #2
RPinPA
Science Advisor
Homework Helper
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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.
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.
- #3
Mark44
Mentor
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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##.
- #4
rugerts
- 153
- 11
Ahh, I see. Thank youRPinPA said:
FAQ: Find eigenvalues & eigenvectors
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.