I don't quite understand the definition of the Pauli matrices, referring to the Wikipedia entry https://en.wikipedia.org/wiki/Pauli_matrices.
How can the three Pauli matrices together with the identity matrix be a basis for the real vector space $\mathbb{R}$ and for the complex vector space $\mathbb{C}$ at the same time? And what exactly is meant by "real vector space of $2 \times 2$ Hermitian matrices"? Is it a real or a complex vector space?
How could I prove these statements mathematically?
Thanks for any type of help 🙂
Best Answer
Let us consider $M := M_2(\mathbb{C})$, the $\mathbb{C}$-vector space of $2\times 2$ matrices with complex coefficients. The description of the structure of $\mathbb{C}$-vector space can be found in any textbook of linear algebra, I guess.
For each $A \in M$, let us denote $M^*$ the transpose of the matrix obtained by taking the complex conjugate of the coefficients of $M$. The matrix $M$ is called the adjoint of $M$. Consider $H$ the subset of $M$ containing the Hermitian matrices, that is, the matrices that are equal to their adjoint.
Lemma: For all $A,B \in H$, for all $\lambda \in \mathbb{R}$, $A + \lambda B \in H$.
Proof: It is straightforward, using some easy properties of the adjoint.
Lemma: If $A \in H$, and if $A \neq 0$, then $iA \not \in H$.
Proof: Also straightforward: assume both $A$ and $iA$ are elements of $H$. Then $(iA)^* = -iA^* = -iA = -(iA)^*$. So $(iA)^* = -(iA)^*$, so $(iA)^* = 0$, so $iA = 0$, so $A = 0$.
Conclusion: $H$ is a real vector subspace, but not a complex vector subspace.
Theorem:
Proof: