I am studying linear algebra by myself, using Serge Lang's "Linear Algebra", and I'm having some difficuties undersending the relationship between transpose, symmetric, Hermitian and unitary operators and their matrices. I have a few questions but, seeing as they all seem very closely related, I decided to post them together. Here they are:
1) Let $V$ be an n-dimensional vector space over $\mathbb{R}$, let $\langle\cdot,\cdot\rangle$ be a scalar product defined on $V$, and let $A:V\rightarrow V$ be a linear operator$^{[1]}$. Moreover, let $\mathcal{B} = \{v_1,v_2,\dots,v_n\}$ be a base of $V$, and $[L]^{\mathcal{B}}_{\mathcal{B}}$ be the matrix representing a linear application $L$ with respect to $\mathcal{B}$. What conditions (non-degenerate, definite positive, etc) are needed on $\langle\cdot,\cdot\rangle$ and/or on $\mathcal{B}$, to make sure that, for each possible choice of $A$, the following hold:
- $[^tA]^{\mathcal{B}}_{\mathcal{B}}=([A]^{\mathcal{B}}_{\mathcal{B}})^T$, where $^tA$ is the only operator such that $\forall v,w \in V, \langle A(v),w\rangle = \langle v,^tA(w)\rangle$
- $^tA=A \iff [^tA]^{\mathcal{B}}_{\mathcal{B}}=([A]^{\mathcal{B}}_{\mathcal{B}})^T$
- $A$ is unitary (that is, $\langle A(v),A(w)\rangle = \langle v,w\rangle$) $\iff [A]^{\mathcal{B}}_{\mathcal{B}}*([A]^{\mathcal{B}}_{\mathcal{B}})^T=I$
2) What would change if, instead of on $\mathbb{R}$, we defined $V$ on any other field $\mathbb{K}$?
3)Let $V$ now be defined on $\mathbb{C}$ and $\langle\cdot,\cdot\rangle$ be an Hermitian product. Same as before, what conditions (non-degenerate, definite positive, etc) are needed on $\langle\cdot,\cdot\rangle$, or possibly on $\mathcal{B}$, to make sure that, for each possible choice of $A$, the following hold:
- $[A^*]^{\mathcal{B}}_{\mathcal{B}}=\overline{([A]^{\mathcal{B}}_{\mathcal{B}})}^T$, where $A^*$ is the only operator such that $\forall v,w \in V, \langle A(v),w\rangle = \langle v,A^*(w)\rangle$
- $A^*=A \iff [^tA]^{\mathcal{B}}_{\mathcal{B}}=\overline{([A]^{\mathcal{B}}_{\mathcal{B}})}^T$
- $A$ is complex unitary (that is, $\langle A(v),A(w)\rangle = \langle v,w\rangle$) $\iff [A]^{\mathcal{B}}_{\mathcal{B}}\ \overline{([A]^{\mathcal{B}}_{\mathcal{B}})}^T=I$
4) What would change if $V$ were not finite dimensional?
I admittedly found some similar questions here on stackexchange, but most of them seem to limit themselves to the standard dot product on $\mathbb{R^n}$ or the standard Hermitian product on $\mathbb{C^n}$, whereas I'm interested in the generic version. Working by myself I found that most of these seem to hold if the matrix $M$ representing the scalar or Hermitian product with respect to $\mathcal{B}$ is either $I$ or $-I$ (which, in a way, makes them isomorphic to the standard dot and Hermitian products), and the implications from matrix equalities to operator equalities also seem to hold if $M$ is something akin to: $$\left(\begin{matrix} \pm I & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & 0\\ \end{matrix}\right).$$ but I'm really not sure if there are other cases which work.
I'd be grateful even for a partial answer, seeing as my post is quite long.
$^{[1]}$: The definition of operator I've learned up until now is simply "linear funcion from a space into itself", so there might be something I'm missing