What is a tensor representation of a 3x3 diagonal matrix with x, y and z in its diagonals respecting the Einstein summation convention?

441 Views Asked by At

What is a good tensor representation of $\left(\begin{array}{c c c}x&0&0\\0&y&0\\0&0&z\end{array}\right)$ where $\vec{r}=(x,y,z)$? (The Einstein summation convention must be followed whenever there are repeated indices.) [For example, the $3\times3$ identity matrix can be represented by $\delta_{\mu\nu}$, $\vec{r}$ can be represented by $r_\mu$, the scalar $r^2$ can be represented by $r_\mu r_\mu$, and so on. Warning: $\delta_{\mu\nu}r_\mu$ does not work since (after summing over $\mu$) it is a vector and not a $3\times3$ object.]

1

There are 1 best solutions below

0
On

This might work $$ a_{ij} = x \delta_{1i}\delta_{1j} + y \delta_{2i}\delta_{2j} + z \delta_{3i}\delta_{3j} $$