# Unitarily Equivalent Operators

• LikeMath
In summary, the conversation discusses the concept of unitary equivalence in matrices and its relationship to eigenvectors. This is defined as the existence of a unitary matrix P such that P^{-1}AP = B. The conversation also mentions that two matrices are considered similar if there exists any matrix P such that AP = PB. It is noted that two matrices are unitarily equivalent if and only if they have the same eigenvectors, which means they represent the same linear transformation in different bases. The speaker then asks if this concept can be generalized to the operator case instead of matrices.

#### LikeMath

Hi!
Could anyone please tell me the meaning of
Tow operators are unitary equivalent.
I tried Wiki but I did not get my goal.

On this page: http://en.wikipedia.org/wiki/Similar_matrix, about "similar matrices", Wikipedia says that two matrices, A and B, are "unitarily equivalent" if and only if there exist a unitary matrix, P, such that $P^{-1}AP= B$. Two matrices are "similar" if there exist any matrix, P, such that $AP= PB$, with P not necessarily unitary. Two matrices are unitarily equivalent if and only if they have the same eigenvectors. That is the same as saying they represent the same linear transformation in different bases.

Thank you, But I wonder if this can be generalized to the operator case instead of matrices

## 1. What are unitarily equivalent operators?

Unitarily equivalent operators are operators in linear algebra that have the same eigenvalues and eigenvectors, but differ only by a change of basis. This means that they represent the same linear transformation, but in different coordinate systems.

## 2. How do unitarily equivalent operators relate to unitary matrices?

Unitarily equivalent operators are closely related to unitary matrices, as they are both defined by the property of preserving inner products. In fact, a unitarily equivalent operator is simply the matrix representation of a linear transformation with respect to a different basis.

## 3. What is the significance of unitarily equivalent operators?

Unitarily equivalent operators are important in quantum mechanics and other areas of physics, where they represent different ways of looking at the same physical system. They also have applications in signal processing and data compression, where they can be used to simplify and analyze complex systems.

## 4. How can unitarily equivalent operators be identified?

Unitarily equivalent operators can be identified by finding a unitary matrix that transforms one operator into the other. This can be done by finding a basis of eigenvectors for each operator and then constructing a unitary matrix from the corresponding eigenvectors.

## 5. Can unitarily equivalent operators be used interchangeably?

Yes, unitarily equivalent operators can be used interchangeably in most cases, as they represent the same linear transformation. However, there may be situations where one representation is more useful or convenient than the other, depending on the context of the problem.