My questions pertain to the following task:
"A Matrix $A \in M(n, \mathbb{R})$ defines a linear map:
$b_A : \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}, \qquad b_A(v,w) := v^{t}Aw$.
For which of the following matrices $A \in M(2, \mathbb{R})$ is the map $b_{A_{i}}$ a scalar product? I.e the map has the properties that it is symmetrical, bilinear and positive definite?
$A_1 = \begin{smallmatrix} 4 & 2 \\ 1 & 4 \end{smallmatrix}, A_2 = , ... , A_6 = ... .$ (Here I have left out several matrices as I don't think they contribute to the question & I intend to check them myself )."
I am not well versed in mathematical notation, & I am finding this question a bit unclear. I interpret the task to mean that I will have shown that $b_{A_{i}}$ is a 'scalar product' if i demonstrate the other 3 properties.
could someone please help me understand what i need to do for this task, so i can complete it for the remaining matrices $A_2 - A_6$?
Every required property of this map, to be a scalar product, corresponds to a certain similar one of the matrix $A$. For example, to show that $b_A$ is symmetric, $$b_A(u,v)=u^tAv\\b_A(v,u)=v^tAu$$ should be equal as scalars. But
$$(v^tAu)^t=u^tA^tv.$$ Thus it is equivalent to $A=A^t$ i.e., $A$ be symmetric. So, to prove this property it is enough to check that $A$ is symmetric.
This map also inherits the linearity from matrix multiplication and distributive laws(you can easily check it).
For positivity, a symmetric matrix is positive definite if all of its eigenvalues are real and positive. To check it you may alternatively take a general vector $x$ and evaluate $c=x^tAx$. The matrix $A$ is positive definite whenever $c$ is positive.