$n$ dimensional matrix from rank $2$ tensors

Question

I just cant understand how can we build $ n $ x $ n$ matrices from a $(0,2)$ and $(2,0)$ tensors' components/coordinates, where $n$ is not $2$. Can someone please give an example ?

Answer 1

A bilinear form is a $(2,0)$ tensor, since it takes two vector inputs, and outputs a scalar. So for example, define a bilinear form on $\Bbb{R}^3$ defined by

$$ B(x,y) = x_1 y_1 + 3 x_2 y_3 - 5x_3 y_3 $$

Then we can associate to $B$ the $3$-by-$3$ matrix of its coefficients:

$$ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 3 \\ 0 & 0 & -5 \end{pmatrix} $$

In general, for any $n$-by-$n$ matrix $B$, you can defined a $(2,0)$ tensor on $\Bbb{R}^n$ by

$$ B(x,y) = \left< x, By \right> $$

where $\left<-,-\right>$ is the standard inner product.

And similarly, you can define the $(0,2)$ tensor as the sum

$$ \sum_{i,j} b_{ij} ~ e_i \otimes e_j $$

where $e_1,\dots,e_n$ is a basis of $\Bbb{R}^n$.