I know the definition of vector space (i.e. it should be closed under addition and multiplication). I read in a book ('Linear Algebra done right' by Sheldon Axler) that the numbers(complex or real) inside the matrices decides wether the vector space they form will be complex or real.
But my question is, since hermitian matrices have complex numbers inside it, then how can it form a real vector space.
A same question is asked here Show that Hermitian Matrices form a Vector Space
But in the answer, he tells that hermitian matrices are not closed under multiplication, how does it implies that it forms real vector vector space (to this logic, hermitian matrices must not even form a vector space, since they are sometimes complex, and on multiplication by complex number it becomes real?
Best Answer
Maybe the misunderstanding is that you think a real vector space consists of real matrices. But real vector space means a vector space where the scalars (for scalar multiplication) are real.
Therefore, the comment from Angina Seng gives the full answer to your question.