Linear Transformations: Find Eigenvalues & Eigenvectors

In summary: So, the conversation is about determining all eigenvalues and eigenvectors of a linear transformation T defined by T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...,z_1+ ... +z_n), where n is a positive integer. The person asking for help has not provided any information about what they have tried so far, making it difficult to give suggestions or hints. However, it is suggested to find the eigenvalues and eigenvectors by writing T as a matrix. Since all components of T(z) are the same, at least one eigenvalue should be obvious. It is also unclear what is meant by "T
  • #1
mivanova
7
0
Please, help me!
Suppose n is a positive integer and T is in F^n is defined by
T(z_1, z_2, ... , z_n) = (z_1+ ... +z_n, z_1+ ... +z_n, ...,z_1+ ... +z_n)
Determine all eigenvalues and eigenvectors of T.
Thank you in advance!
 
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  • #2
Thanks in advance for what? Telling you the answer? No, that's not going to happen.

Making suggesteins and giving you hints? Hard to do without knowing what you DO know about this problem. And since you have shown nothing at all about what you have tried, we can't know that. Do you, for example, know how to write T as a matrix? Could you find the eigenvalues and eigenvectors then? Since all of the components of the T(z) are the same, no matter what z is, at least one eigenvalue should be obvious. What is T(1, -1, 0, 0, ..., 0)?

I am wondering what, exactly, you mean by "T is in F^n". My first guess would be that F^n is the set of ordered n-tuples of some field F, with component wise addition and scalar multiplication. But in that case it is (z_1, z_2, ..., z_n) that is "in F^n", not T. T is in L(F^n, F^n).
 

FAQ: Linear Transformations: Find Eigenvalues & Eigenvectors

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the basic structure of the original space. This means that the transformation preserves the operations of addition and scalar multiplication.

2. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are important concepts in linear algebra that are used to describe the behavior of a linear transformation. An eigenvector is a vector that, when transformed by a linear transformation, remains parallel to its original direction. The corresponding eigenvalue is the scalar that scales the eigenvector during the transformation.

3. How do you find eigenvalues and eigenvectors?

To find eigenvalues and eigenvectors, you need to solve the characteristic equation of the linear transformation. This equation is determined by finding the values of the transformation that, when multiplied by the original vector, result in a scalar multiple of the original vector. These values are the eigenvalues, and the corresponding eigenvectors can be found by solving the system of equations formed by the characteristic equation.

4. Why are eigenvalues and eigenvectors important in linear transformations?

Eigenvalues and eigenvectors provide important information about the behavior of a linear transformation. They can help to identify the direction and magnitude of the transformation, as well as any fixed or invariant points in the transformation. They are also useful in applications such as computer graphics, data analysis, and quantum mechanics.

5. Can a linear transformation have more than one set of eigenvalues and eigenvectors?

Yes, a linear transformation can have multiple sets of eigenvalues and eigenvectors. This is because there can be multiple eigenvectors that correspond to the same eigenvalue. In this case, the set of eigenvectors form a subspace called the eigenspace, and the eigenvalue is referred to as the multiplicity of the eigenspace.

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