Matrices and quadratic forms: meaning of a symbolism

Question

Definition: The $A$ matrix is said to be positive if it is $A_{ij} x^i x^j > 0 \quad \forall \; x^i \neq 0^i$

I know $A$ is a matrix: but what about $i$ and $j$ indexes and $x^i$ $x^j$?

Answer 1

Using Einstein's summation convention,

$$A_{ij} x^i x^j$$ is a shorthand for the sum

$$\sum_{i,j}A_{ij}x_ix_j=x^TAx.$$

https://en.wikipedia.org/wiki/Einstein_notation

Superscripts are used rather than subscripts because the convention requires "covariant" vs. "contravariant" components. The concepts generalizes to higher order tensors.

Answer 2

Assume that $A$ is a $2\times2$ matrix (for the sake of simplicity), and $X$ as a $2\times 1$ vector; the matrix $A$ is said to be positive if we have that:

$$X^TA X >0$$

if you try writing down this example you will realise from where you obtain the expression you have (there's a missing summation though)

The indexes $i,j$ over the $x$ just make reference to the $ith$ and the $jth$ components of the vector $X$.

(of course that if $X$ is the null vector, the condition won't be satisfied even if A is positive)