# Positive non self adjoint operator?

• broncos#1
In summary, a positive non-self adjoint operator is a linear transformation between two vector spaces that is not equal to its adjoint and maps positive vectors to positive vectors. It differs from a positive self-adjoint operator in that it lacks the symmetry property and may have complex eigenvalues. These operators have various applications in mathematics, physics, and engineering. They are a special case of Hermitian operators, which have real eigenvalues and additional properties. Positive non-self adjoint operators cannot have negative eigenvalues, as they are defined to only map positive vectors to positive vectors.
broncos#1

## Homework Statement

Find an operator T on F^2 such that <T(v),v> is greater than or equal to 0 for all v in F^2, but T is not self-adjoint

## The Attempt at a Solution

I am trying to to find examples where <T(v),v>=0 where the operator is not 0

broncos#1 said:
I am trying to to find examples where <T(v),v>=0 where the operator is not 0
Can't you use algebra to solve for all possible values of T?

## 1. What is a positive non-self adjoint operator?

A positive non-self adjoint operator is a mathematical object that represents a linear transformation between two vector spaces, where the operator is not equal to its adjoint. Additionally, the operator is positive, which means that it maps positive vectors to positive vectors.

## 2. How is a positive non-self adjoint operator different from a positive self-adjoint operator?

A positive non-self adjoint operator differs from a positive self-adjoint operator in that it does not possess the symmetry property of being equal to its adjoint. This means that the operator maps vectors to a different space than its adjoint does, and it may also have complex eigenvalues.

## 3. What are some common applications of positive non-self adjoint operators?

Positive non-self adjoint operators have many applications in mathematics, physics, and engineering. They are used to model phenomena such as heat flow, fluid dynamics, and quantum mechanics. They are also used in numerical analysis to solve differential equations and in optimization problems.

## 4. How are positive non-self adjoint operators related to Hermitian operators?

Positive non-self adjoint operators are a special case of Hermitian operators, which are self-adjoint and have real eigenvalues. However, positive non-self adjoint operators may have complex eigenvalues and are not equal to their adjoints. Hermitian operators also have additional properties, such as orthogonality of eigenvectors.

## 5. Can positive non-self adjoint operators have negative eigenvalues?

No, positive non-self adjoint operators are defined as operators that map positive vectors to positive vectors. This means that they can only have positive or zero eigenvalues. If a positive non-self adjoint operator has a negative eigenvalue, it would violate this definition and would instead be classified as a negative operator.

• Calculus and Beyond Homework Help
Replies
2
Views
3K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
552
• Calculus and Beyond Homework Help
Replies
24
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
497
• Calculus and Beyond Homework Help
Replies
0
Views
596
• Calculus and Beyond Homework Help
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
621
• Calculus and Beyond Homework Help
Replies
15
Views
1K
• Calculus and Beyond Homework Help
Replies
16
Views
2K