# Difference symmetric matrices vector space and hermitian over R

## Main Question or Discussion Point

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?

## Answers and Replies

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fresh_42
Mentor
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 $z\in \mathbb{R}$.