Eigenvectors of Matrix: Solving Basics

In summary, the conversation is about finding the eigenvectors of a given matrix and whether a particular solution is valid. The solution is confirmed to be valid as any scalar multiple of it is also a valid solution. The concept of eigenvectors forming a subspace is also mentioned.
  • #1
daveyman
88
0

Homework Statement



Find the eigenvectors of the following matrix:

[tex]
\[ \left( \begin{array}{ccc}
1 & 1 \\
4 & -2 \end{array} \right)\]
[/tex]

Homework Equations



N/A

The Attempt at a Solution



I already know how to find the solutions. They are {1 1} and {-1 4}. My question is this: could a solution also be {1 -4} (switching the minus sign)? I don't think it is a solution but I don't see why not...

Any help would be great. Thanks!
 
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  • #2
Have you tried [tex]\begin{pmatrix}1 \\ -4\end{pmatrix}[/tex] ? All it takes to be a solution is solving the eigenvalue problem for some eigenvalue [tex]\lambda[/tex].

Tom
 
  • #3
your sol'n times any c in R is fine
 
  • #4
davyjones said:
your sol'n times any c in R is fine

Oh I guess that's obvious huh. So if c=-1, I'm good.

Okay thanks!
 
  • #5
Exactly.
 
  • #6
One important thing you should have learned (or maybe this exercise is in preparation) the set of all eigenvectors (corresponding to a given eigenvalue) forms a subspace: it is closed under both vector addition and scalar multiplication.
 

FAQ: Eigenvectors of Matrix: Solving Basics

What is an eigenvector?

An eigenvector is a vector that is not changed in direction by a given linear transformation. In the context of matrices, an eigenvector is a vector that, when multiplied by a particular matrix, results in a scalar multiple of itself.

How do I find the eigenvectors of a matrix?

To find the eigenvectors of a matrix, you need to first find the eigenvalues of the matrix. Then, for each eigenvalue, solve the equation (A - λI)x = 0, where A is the matrix, λ is the eigenvalue, and x is the eigenvector. The non-zero solutions to this equation will be the eigenvectors.

What is the importance of eigenvectors?

Eigenvectors are important in many areas of mathematics and science, including linear algebra, physics, and computer science. They are used to understand and analyze linear transformations, diagonalize matrices, and solve differential equations.

Can a matrix have more than one eigenvector?

Yes, a matrix can have multiple eigenvectors for the same eigenvalue. In fact, there can be an infinite number of eigenvectors for a given eigenvalue. This is because any scalar multiple of an eigenvector is also an eigenvector.

How are eigenvectors and eigenvalues related?

Eigenvectors and eigenvalues are related in the sense that eigenvectors are the vectors associated with eigenvalues. Every eigenvalue has at least one corresponding eigenvector, and vice versa. The eigenvalues and eigenvectors of a matrix provide important information about its properties and behavior.

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