# Operator acting in orthogonal subspace

• seek
In summary, for a normal operator A, if y exists in Rperpendiculara1, then Ay also exists in Rperpendiculara1. This can be proven using the fact that all normal operators are linear and satisfy the equation A*A* = A*A. Therefore, A*y must also be in Rperpendiculara1, making Ay an eigenvector of A with eigenvalue a1.
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## Homework Statement

Consider a normal operator A

If Rperpendiculara1 is the orthogonal complement to the subspace of eigenvectors of A with eigenvalue a1, show that if y exists in all Rperpendiculara1 then Ay exists in all Rperpendiculara1

## The Attempt at a Solution

This could be answered very simply, if I knew that all normal operators were linear. Were that the case, A would simply be mapping y onto the subspace in which it already existed. Am I right?

*Edit:
Wikipedia tells me that a normal operator is indeed a linear operator. I think I can do this now.

Last edited:
Let y be a vector in Rperpendiculara1. We know that for any normal operator A, it satisfies the equation A*A* = A*A. Thus, A*y must also be in Rperpendiculara1, since it is an eigenvector of A with eigenvalue a1. Therefore, Ay exists in all Rperpendiculara1.

## 1. What is an operator acting in an orthogonal subspace?

An operator acting in an orthogonal subspace is a mathematical concept that relates to linear transformations. An operator is a function that maps one vector to another vector, while an orthogonal subspace is a vector space that is perpendicular to another vector space. Therefore, an operator acting in an orthogonal subspace is a function that transforms vectors in a particular subspace that is perpendicular to the original vector space.

## 2. How is an operator acting in an orthogonal subspace different from a regular operator?

An operator acting in an orthogonal subspace differs from a regular operator in that it only affects vectors in a specific subspace that is perpendicular to the original vector space. A regular operator, on the other hand, can affect vectors in any direction within the original vector space.

## 3. What is the significance of an operator acting in an orthogonal subspace?

An operator acting in an orthogonal subspace is significant in linear algebra because it allows for the separation of a vector space into subspaces that are independent of each other. This makes it easier to analyze and understand the behavior of linear transformations on specific subspaces.

## 4. How is an operator acting in an orthogonal subspace used in practical applications?

An operator acting in an orthogonal subspace has various practical applications, especially in fields that involve vector spaces and linear transformations. For example, it is used in signal processing to filter out specific frequencies, in computer graphics to rotate objects, and in quantum mechanics to represent the spin of a particle.

## 5. Can an operator acting in an orthogonal subspace be represented by a matrix?

Yes, an operator acting in an orthogonal subspace can be represented by a matrix. This is because an operator acting in an orthogonal subspace is essentially a linear transformation, and matrices are commonly used to represent linear transformations in mathematics. The matrix representation of an operator acting in an orthogonal subspace will have special properties, such as being an orthogonal matrix.

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