Hermitian matrix vector space over R proof

In summary, the conversation discusses proving that a Hermitian matrix is a vector space over R. The equations and attempt at a solution are provided, including the knowledge that the sum of two Hermitian matrices and multiplication by a real number result in a Hermitian matrix. The expert clarifies that complex numbers are not elements of R and that the proof provided is sufficient.
  • #1
Jimena29
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0

Homework Statement


I need to prove that the hermitian matrix is a vector space over R

Homework Equations





The Attempt at a Solution


I know the following:
If a hermitian matrix has aij = conjugate(aji) then its easy to prove that the sum of two hermitian matrices A,B give a hermitian matrix.
Multiplying a Hermitian matrix by a real number k will also give a hermitian matrix.
The zero vector is an element of the set of all Hermitian matrices.
So I can prove that hermitian matrices are a vector space, but what I`m stuck on is how a hermitian matrix is an element of real matrices??
How are complex numbers elements of R?
Thank you!
 
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  • #2
A complex number is not an element of R, nor does a Hermitian matrix necessarily have real elements. You've already proved exactly what you need to prove. The Hermitian matrices are a vector space over R. They wouldn't be a vector space over C.
 
  • #3
Oh, I thought I also had to prove that the matrix A is contained in R
Thank you very much!
 

1. What is a Hermitian matrix vector space over R?

A Hermitian matrix vector space over R is a vector space in which the inner product of any two vectors is equal to the complex conjugate of the inner product of the second vector with the first. In other words, it is a vector space with a symmetric and positive definite inner product.

2. How do you prove that a matrix is Hermitian?

A matrix is Hermitian if it is equal to its own conjugate transpose. This can be proven by showing that the matrix is equal to its conjugate transpose, and then using the properties of the inner product to show that the inner product of any two vectors in the matrix is equal to the complex conjugate of the inner product of the second vector with the first.

3. What are the properties of a Hermitian matrix vector space over R?

Some properties of a Hermitian matrix vector space over R include: the inner product is symmetric, positive definite, and linear in the first argument; the inner product of a vector with itself is always non-negative; and the inner product of two orthogonal vectors is always equal to zero.

4. How does the Hermitian property affect diagonalization of a matrix?

The Hermitian property of a matrix allows it to be easily diagonalized. This means that the matrix can be transformed into a diagonal matrix with only real values on the diagonal, which makes it easier to perform calculations and solve problems involving the matrix.

5. What are some applications of Hermitian matrix vector spaces over R?

Hermitian matrix vector spaces over R have many applications in mathematics, physics, and engineering. They are commonly used in quantum mechanics, signal processing, and control theory. They are also used in optimization problems and in the analysis of data, such as in principal component analysis.

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