# Derivative of a vector with respect to a matrix

I am at an impasse. I don't know if homework is allowed on here or not, so if it isn't, someone delete this.

Given:

$H_{\gamma} = C_{\beta \beta} v_{\gamma} + C_{\beta \varepsilon} C_{\varepsilon \beta} v_{\gamma} + C_{\beta \varepsilon} v_{\varepsilon} C_{\beta \gamma} + \delta_{\beta \gamma} v_{\beta} + \varepsilon_{\gamma \epsilon \eta} v_{\varepsilon} v_{\eta} + \varepsilon_{\gamma \varepsilon \eta} C_{\varepsilon \eta}$

Find:

$\frac{\partial H_{\gamma}}{\partial C_{\alpha \beta}}$

I believe $\delta$ is the identity matrix, and $\varepsilon$ is the Levi-Civita symbol. I don't know how to work this sort of thing and my notes aren't helping. I have one example that takes the partial of a scalar function with respect to a matrix, but this is confusing me.

Does the $\beta$ in the definition of $H_{\gamma}$ refer to the same $\beta$ in $\frac{\partial H_{\gamma}}{\partial C_{\alpha \beta}}$?

I can't even tell what kind of tensor this result is. Is my result a matrix, a vector, or a vector of matrices? If someone could maybe walk me through even the first term ($\frac{\partial C_{\beta \beta} v_{\gamma} }{\partial C_{\alpha \beta}}$) that would be very helpful.

My textbook doesn't include this sort of notation (only my lecture notes), so that's not very helpful. And Googling "derivative of a vector with respect to a matrix" is useless.

Thanks!

-
are you using Einstein notation where a repeated index is summed over? –  Jonathan Oct 9 '12 at 4:26
Yes, I am using Einstein notation. –  Nick Oct 9 '12 at 4:57