# Lagrangian Multiplier with Matrices

1. Sep 17, 2009

### CuppoJava

Hi,
I'm trying to use calculus of variations to solve for the probability distribution with highest entropy for a given covariance matrix. I want to maximize this:

$$H[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x}$$

with the following constraints:

$$\int p(\vec{x}) = 1$$
$$\int \vec{x}p(\vec{x})d\vec{x} = \vec{u}$$
$$\int (\vec{x}-\vec{u})(\vec{x}-\vec{u})^{T}p(\vec{x})d\vec{x} = \Sigma$$

Using Lagrangian multipliers, the proposed maximization function is:

$$F[p(\vec{x})] = -\int p(\vec{x})*ln(p(\vec{x}))d\vec{x} + \lambda_{1}(\int p(\vec{x})d\vec{x}-1) + \vec{m}^{T}(\int \vec{x}p(\vec{x})d\vec{x} - \vec{u}) + Tr\{L(\int (\vec{x}-\vec{u})(\vec{x}-\vec{u})^{T}p(\vec{x})d\vec{x} - \Sigma)\}$$

I understand that m is needed because there is D constraints imposed by the mean. And L is needed because there is DxD constraints imposed by the covariances. But what is the trace operator doing in there?

Thanks for the help
-Patrick