Difference symmetric matrices vector space and hermitian over R

In summary: Therefore, the two questions are asking for the proof of vector space properties for different types of matrices, one being a subspace of the other.
  • #1
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Hi guys,
I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars)

However, then I was asked to prove that the space of hermitian matrices is a vector space over R. I fail to see the difference between the two questions, as I thought hermitian matrices over R did not have any complex entries and therefore were just regular symmetric matrices.

Can anyone enlighten me as to what the difference between these two questions are?
 
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  • #2
Rowina said:
Hi guys,
I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars)

However, then I was asked to prove that the space of hermitian matrices is a vector space over R. I fail to see the difference between the two questions, as I thought hermitian matrices over R did not have any complex entries and therefore were just regular symmetric matrices.

Can anyone enlighten me as to what the difference between these two questions are?
Hermitian matrices ##\overline{A}=A^\tau ## do have complex entries, just not at the diagonal, and the rest are complex conjugates between upper and lower triangular submatrices. The clue here is, that they do not build a complex vector space, because ##\overline{z \cdot w} \neq z \cdot \overline{w}##, but a real vector space, because ##\overline{z}=z ## for ##z\in \mathbb{R}##.
 

What are difference symmetric matrices vector space and hermitian over R?

Difference symmetric matrices vector space refers to the set of all matrices that are symmetric and have a zero diagonal. Hermitian over R refers to the set of all matrices that are equal to their own conjugate transpose.

What is the difference between symmetric and hermitian matrices?

The main difference is that symmetric matrices are defined over real numbers, while hermitian matrices are defined over complex numbers. Additionally, hermitian matrices have the property that they are equal to their own conjugate transpose, while symmetric matrices have the property that they are equal to their own transpose.

How are symmetric and hermitian matrices related to vector spaces?

Both symmetric and hermitian matrices are subspaces of the vector space of all matrices. This means that they follow the same rules and properties as other vector spaces, such as closure under addition and scalar multiplication.

What are some real-world applications of symmetric and hermitian matrices?

Symmetric and hermitian matrices are widely used in fields such as physics, engineering, and computer science. They are particularly useful for solving systems of linear equations, as well as for representing and analyzing data in a variety of applications.

How can one determine if a matrix is symmetric or hermitian?

To determine if a matrix is symmetric, you can check if it is equal to its own transpose. To determine if a matrix is hermitian, you can check if it is equal to its own conjugate transpose. In both cases, if the matrix satisfies these conditions, it can be classified as either symmetric or hermitian.

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