# Eigenvalue of Sum of Eigenvectors

• TranscendArcu
In summary, the conversation involves determining whether the statement T(X+Y) = T(X) + T(Y) is true for different values of the eigenvalues λ_1 and λ_2. The speaker initially shows that it is true when λ_1 = λ_2, but has doubts about the case when they are not equal. They are then encouraged to try to show that it is not true and are given an example to consider. The final question asks for the corresponding eigenvectors and the result of multiplying the given matrix by their sum.
TranscendArcu

## The Attempt at a Solution

So, first I wrote,

$T(X) = λ_1 X, T(Y) = λ_2 Y$

If $λ_1 = λ_2$:

$T(X+Y) = T(X) + T(Y) = λ_1 X + λ_2 Y = λ_1 (X+Y)$,

so this does indeed seem to be an eigenvector. But I'm less convinced for the case $λ_1 ≠ λ_2$. Again, I get the transformation down to the form:

$λ_1 X + λ_2 Y$

But if eigenvalues are necessarily one constant, then I don't see how I pull constants out as above. How do I go about showing this?

Did you read the problem carefully? It doesn't ask you to show that this is true, it asks you to determine whether or not it is true and the show that.

Okay, if you have doubts about this being true, how about trying to show its not true? If it is not, you should be able to show a counterexample.
Obviously
$$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$$
has 1 and 2 as eigenvalues. What are corresponding eigenvalues? What do you get if you multiply that matrix by their sum?

What are corresponding eigenvalues?
I'm pretty sure HallsOfIvy meant "corresponding eigenvectors".

Yes, thanks.

## 1. What is an eigenvalue of the sum of eigenvectors?

An eigenvalue of the sum of eigenvectors is a scalar value that represents the amount by which the sum of eigenvectors is scaled when multiplied by a linear transformation matrix.

## 2. How is the eigenvalue of the sum of eigenvectors calculated?

The eigenvalue of the sum of eigenvectors can be calculated by solving the characteristic equation det(A - λI) = 0, where A is the linear transformation matrix and λ is the eigenvalue.

## 3. Can the eigenvalue of the sum of eigenvectors be negative?

Yes, the eigenvalue of the sum of eigenvectors can be negative. It represents the direction and magnitude of the transformation applied to the eigenvectors.

## 4. How is the eigenvalue of the sum of eigenvectors used in applications?

The eigenvalue of the sum of eigenvectors is used in many applications, such as data analysis, image processing, and signal processing. It helps determine the most important features or patterns in a dataset or image.

## 5. What is the significance of the eigenvalue of the sum of eigenvectors in linear algebra?

The eigenvalue of the sum of eigenvectors is significant in linear algebra because it provides insight into the behavior of linear transformations and their effect on vectors. It also helps solve systems of linear equations and diagonalize matrices.

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