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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?

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    $\begingroup$ Because (i) the sum of Hermitian matrices is Hermitian, (ii) the product of a real number and a Hermitian matrix is a Hermitian matrix. $\endgroup$ – Angina Seng Jul 23 '20 at 4:45
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    $\begingroup$ I am not sure about your question... Matrices are a tool to describe transformations of vector spaces. But it seems that question refers to the vector space consisting of matrices. That is something entirely different. If you think of matrices as description of transformations, then yes, they tell you what field you are looking at. But if you are looking at the vector space of matrices, you still have to look at the matrices which describe the transformations on this space to see what field we are talking about. Please tell if I am correct. Then I can write a more complete answer. $\endgroup$ – Mushu Nrek Jul 23 '20 at 4:45
  • $\begingroup$ I want to use matrices as an operator in quantum mechanics which gives real eigenvalues. But I don't understand why they are real, I want to ask what properties of a matrix differentiate between real and complex matrices? @MushuNrek $\endgroup$ – sawan kt Jul 23 '20 at 4:52
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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.

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  • $\begingroup$ Thankyou I got it $\endgroup$ – sawan kt Jul 23 '20 at 12:54

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