Definition of positive definite matrix

Question

Definitions in the book by Friedberg.

A linear operator $T$ on a finite-dimensional inner product space is called positive definite if $T$ is self-adjoint and $\langle T(x),x\rangle >0$ for all $ \neq 0$. An $n\times n$ matrix $A$ with entries from $\mathbb R $ or $\mathbb C$ is called positive definite if $L_A$ is positive definite.

Question

1. Is the definition of positive definite matrix defined with respect to the dot product of $\mathbb R $ or $\mathbb C$?

2. Do the definition of positive definite matrix depends on what inner product of $\mathbb R $ or $\mathbb C$ is given?

Thanks.

Answer 1

Yes and no. A matrix $A \in M_n(\mathbb{F})$ (where $\mathbb{F} = \mathbb{R}$ or $\mathbb{F} = \mathbb{C}$) is called positive definite if the operator $L_A \colon \mathbb{F}^n \rightarrow \mathbb{F}^n$ is a positive definite operator when $\mathbb{F}^n$ is endowed with the standard inner product. It might be the case that the operator $L_A$ is positive definite as an operator with respect to a different inner product on $\mathbb{F}^n$ but then you won't say that the matrix $A$ is positive definite (unless it is also positive definite as an operator with respect to the standard inner product).