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I develop open-source statistical software (http://openmx.psyc.virginia.edu/), but matrix calculus is not my strong point. I need the 1st and 2nd derivatives of the log multivariate normal density. I was happy to find the 1st derivatives here on CrossValidated,

How to take derivative of multivariate normal density?

However, the 2nd derivatives are left as an exercise to the reader. I am sure the 2nd derivatives have been independently derived many times. However, I cannot find them exhibited anywhere. Is there a calculus expert out there who can detail the 2nd derivatives?

Many thanks.

I develop open-source statistical software (http://openmx.psyc.virginia.edu/), but matrix calculus is not my strong point. I need the 1st and 2nd derivatives of the log multivariate normal density. I was happy to find the 1st derivatives here on CrossValidated,

How to take derivative of multivariate normal density?

However, the 2nd derivatives are left as an exercise to the reader. I am sure the 2nd derivatives have been independently derived many times. However, I cannot find them exhibited anywhere. Is there a calculus expert out there who can detail the 2nd derivatives?

Many thanks.

I develop open-source statistical software (http://openmx.psyc.virginia.edu/), but matrix calculus is not my strong point. I need the 1st and 2nd derivatives of the log multivariate normal density. I was happy to find the 1st derivatives here on CrossValidated,

How to take derivative of multivariate normal density?

However, the 2nd derivatives are left as an exercise to the reader. I am sure the 2nd derivatives have been independently derived many times. However, I cannot find them exhibited anywhere. Is there a calculus expert out there who can detail the 2nd derivatives?

Many thanks.

Alright, the negative information matrix for $L(\mu,K)$ is

$$\frac{\partial^2 L}{\partial\mu\partial\mu} = -\left(\frac{1}{N}K\right)^{-1}$$

$$\frac{\partial^2 L}{\partial K\partial K} = -\left(\frac{2}{N} K \otimes K\right)^{-1}$$

$$\frac{\partial^2 L}{\partial K\partial \mu} = 0$$

However, $\frac{\partial^2 L}{\partial K\partial K}$ as given above is the wrong shape for the multivariate case. For factors=3, it should be square with size 3*(3-1)/2 but it is size factors^2 instead. It seems like the numbers are correct, but I am not sure about the pattern to reduce the matrix to the correct dimensions.

I develop open-source statistical software (http://openmx.psyc.virginia.edu/), but matrix calculus is not my strong point. I need the 1st and 2nd derivatives of the log multivariate normal density. I was happy to find the 1st derivatives here on CrossValidated,

How to take derivative of multivariate normal density?

However, the 2nd derivatives are left as an exercise to the reader. I am sure the 2nd derivatives have been independently derived many times. However, I cannot find them exhibited anywhere. Is there a calculus expert out there who can detail the 2nd derivatives?

Many thanks.