Lin. Algebra - Find Eigenvectors / eigenvalues

In summary, the problem asks to find all eigenvalues and eigenvectors of T in L(F3), where T is defined as (2*z2, 0, 5*z3). By setting up a system of equations, we find that the eigenvalues are 0 and 5. The corresponding eigenvectors are (1, 0, 0) and any multiple of (0, 0, 1), respectively.
  • #1
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Homework Statement


Define T in L(F3) by T(z1, z2, z3) = (2*z2, 0, 5*z3). Find all eigenvalues and eigenvectors of T.

Homework Equations


The Attempt at a Solution


Well, since we want to find all the eigenvalues, we want the following equation to hold:
T(z1, z2, z3) = (2*z2, 0, 5*z3) = [tex]\lambda[/tex](z1, z2, z3).

Setting up a system of equations, I get the following:

[tex]\lambda[/tex]z1 = 2*z2
[tex]\lambda[/tex]z2 = 0
[tex]\lambda[/tex]z3 = 5*z3

Solving, I get one eigenvalue, namely 5. Then I came up with the corresponding eigenvector, (0, 0, 5). Is this correct?

Edit: 0 is the other eigenvalue, but I don't know what the corresponding eigenvector would be.
 
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  • #2
From the second equation, either [itex]\lambda= 0[/itex] or z2= 0. If [itex]\lambda[/itex] is not 0, the z2= 0, and so z1= 0. The last equation is [itex]\lambda[/itex]z3= 5z3 so either z3= 0 or [itex]\lambda= 5[/itex]. Yes [itex]\lambda[/itex]= 5 is an eigenvalue. That gives no restriction on z3 at all. Any multiple of (0, 0, 1) is an eigenvector.

But don't forget the other possibility. If [itex]\lambda[/itex]= 0 then 5z3= 0 so z3= 0. Also 2z2= 0 so z2= 0. But that gives no restriction on z1! For any z1, T(z1, 0, 0)= (0, 0, 0) so 0 is an eigenvalue with any multiple of (1, 0, 0) as eigenvector.
 
  • #3
Thanks for making things more clear. That helped a lot.
 

1. What is an eigenvector and eigenvalue?

An eigenvector is a vector that, when multiplied by a given matrix, results in a scalar multiple of itself. The corresponding scalar value is known as the eigenvalue.

2. Why are eigenvectors and eigenvalues important in linear algebra?

Eigenvectors and eigenvalues are important because they can help simplify complex matrix operations, such as matrix diagonalization and finding the inverse of a matrix. They also have many applications in physics, engineering, and data analysis.

3. How do you find eigenvectors and eigenvalues?

To find eigenvectors and eigenvalues, you first need to set up and solve a characteristic equation. This involves finding the determinant of the matrix and solving for the roots of the equation. The resulting eigenvalues can then be used to find the corresponding eigenvectors.

4. What are some real-world applications of eigenvectors and eigenvalues?

Eigenvectors and eigenvalues have many applications in fields such as physics, engineering, and data analysis. For example, they are used in quantum mechanics to describe the energy states of particles, in image processing to identify patterns and features, and in network analysis to identify important nodes.

5. Can a matrix have multiple eigenvectors and eigenvalues?

Yes, a matrix can have multiple eigenvectors and eigenvalues. In fact, most matrices have multiple eigenvectors and eigenvalues. The number of eigenvectors and eigenvalues is equal to the dimension of the matrix.

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