# ST and TS have the same eigenvals

• balletomane
In summary, the conversation discusses proving that for linear operators S and T on a vector space V, ST and TS have the same eigenvalues. The speaker has reached the point of showing that TS has an eigenvalue g corresponding to the eigenvector Tu, but is unsure how to guarantee that Tu is nonzero or how to handle the possibility of Tu being zero. They also question if zero as an eigenvalue is a special case and if the nonzero eigenvalue case needs to be handled separately.
balletomane
Hi.

I need to prove that for S,T linear operators on V. ST and TS have the same eigenvalues. I've gotten as far as (say g is the eigenvalue and u is a nonzero vector): STu=gu so TS(Tu)=g(Tu). So TS has eigenvalue g corresponding to eigenvector Tu. But I don't know how to guarantee that Tu is nonzero. (Or how to resolve the possibility that it is zero)

Thanks for any help.

If Tu is zero, what's STu?

If Tu=0

STu=0=gu, but u is nonzero, so g=0?

So does zero as an eigenvalue become a special case? Then would I do the nonzero eigenvalue case separately?

## 1. What are eigenvalues in relation to ST and TS?

Eigenvalues are a mathematical concept that represents the scalar values that are associated with a linear transformation. In the case of ST and TS having the same eigenvalues, it means that the two linear transformations have the same set of scalar values associated with them.

## 2. How can ST and TS have the same eigenvalues?

ST and TS can have the same eigenvalues if they are commutative, meaning that the order in which they are applied does not affect the result. This is because the eigenvalues are determined by the underlying matrix of a linear transformation, which is unchanged by multiplication order in commutative transformations.

## 3. What does it mean for ST and TS to have the same eigenvalues?

When ST and TS have the same eigenvalues, it means that they share certain characteristics. For example, they may have the same determinant or trace, which are also determined by the eigenvalues. It also means that the two transformations have similar behavior and can be interchanged without changing the overall result.

## 4. Can ST and TS have different eigenvalues?

Yes, ST and TS can have different eigenvalues. This would mean that the two transformations have different underlying matrices and therefore, different characteristics. This could result in different behaviors and results when the transformations are applied.

## 5. What are some applications of ST and TS having the same eigenvalues?

Having the same eigenvalues for ST and TS can be useful in matrix diagonalization, which is a common technique used in data analysis and signal processing. It can also be helpful in finding eigenvectors, which have many applications in fields such as physics, engineering, and computer science.

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