Is cross pruduct hermitian?

Thread starter ber70
Start date Dec 15, 2009
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Cross Hermitian

In summary, a cross product is a mathematical operation between two vectors that results in a third vector perpendicular to the first two. It can be represented as a hermitian matrix or a skew-hermitian matrix. It is important to determine if a cross product is hermitian because it can provide information about the properties of the vectors involved. This can be determined by checking if the cross product of the complex conjugates of the vectors is equal to the original cross product. If a cross product is not hermitian, it may still have other properties but may not be as useful in certain contexts. A cross product cannot be both hermitian and anti-hermitian due to the contradictory properties of these matrices.

Dec 15, 2009

ber70

A and B are two hermitian vector operators.
K1=AXB, K2=AXB-BXA.
Are K1 and K2 hermitian or anti-hermitian?

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Dec 15, 2009

phsopher

Well what do you think?

1. What is a cross product and how is it related to hermitian matrices?

A cross product is a mathematical operation between two vectors that results in a third vector perpendicular to the first two. It is related to hermitian matrices because the cross product of two complex vectors can be represented as a hermitian matrix, and the cross product of two real vectors can be represented as a skew-hermitian matrix.

2. Why is it important to determine if a cross product is hermitian?

Determining if a cross product is hermitian can provide information about the properties of the vectors involved. Hermitian matrices have several important properties, such as being diagonalizable and having real eigenvalues, which can be useful in various applications.

3. How can you tell if a cross product is hermitian?

To determine if a cross product is hermitian, you can use the definition of a hermitian matrix, which states that the conjugate transpose of the matrix must be equal to the original matrix. In the case of a cross product, this means that the cross product of the complex conjugates of the two vectors involved must be equal to the original cross product.

4. What happens if a cross product is not hermitian?

If a cross product is not hermitian, it can still have other properties, such as being anti-hermitian or unitary. However, it may not have the same useful properties as a hermitian cross product, and it may not be as applicable in certain mathematical or scientific contexts.

5. Can a cross product ever be both hermitian and anti-hermitian?

No, a cross product cannot be both hermitian and anti-hermitian. A hermitian matrix has the property that it is equal to its own conjugate transpose, while an anti-hermitian matrix has the property that it is equal to the negative of its conjugate transpose. These two properties are contradictory, so a cross product cannot satisfy both at the same time.