We may have a non-informative prior for a location parameter $u$: $p(u)=\dfrac{1}{L}.$
We may have a non-informative prior for a location parameter of vector $\textbf{u}$ (their components are mutually independent): $p(\textbf{u})=\dfrac{1}{L_1 L_2 \dots L_n}.$
We have a non-informative prior for a scale parameter $t$ (uniform on a log scale): $p(t)=\dfrac{1}{tL}.$
How do you define a non-informative prior for a covariance matrix which is positive definite but not diagonal?
If someone would like to give the Jeffreys prior, please give the full form of it (just shown above).