# Demonstrate the matrix represents a 2nd order tensor

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1. Oct 18, 2015

### Dazed&Confused

1. The problem statement, all variables and given/known data
Demonstrate that matrix $T$ represents a 2nd order tensor

$T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}$
2. Relevant equations

To show that something is a tensor, it must transform by $T_{ij}' = L_{il}L_{jm}T_{lm}$. I cannot find a neat general form for $T_{ij}$
3. The attempt at a solution

This has been difficult to do, so I'm running of options. A hint may be helpful.

2. Oct 18, 2015

### fzero

Try $T_{ij} = - x_i x_j + \delta_{ij} A$. Compute $A$ by comparing the trace of this expression with the trace of that matrix.

3. Oct 19, 2015

### Dazed&Confused

Thanks. The A is going to be $x_kx_k$. This is a scalar, so the transformation looks correct.