= Isotropic second-rank tensor
{title2=$T_{ij}=\lambda\delta_{ij}$}
A second-rank Cartesian <tensor> invariant under every proper <rotation matrix> is a scalar multiple of the <Kronecker delta>. Half-turns about the coordinate axes kill its off-diagonal entries, and quarter-turns make its diagonal entries equal. A tensor of the form $v_iw_j$ is an <outer product> with matrix rank at most one, so it is isotropic in three dimensions only when it vanishes, equivalently when $v=0$ or $w=0$.
Back to article page
