# Least Squares Fit Covariance Matrix

How do you perform a least squares fit for $$\Sigma$$ in the equation $$v = u^T\Sigma u$$? The term $$v$$ is a vector of observed variances of a projected Gaussian distribution. The matrix $$u$$ is made of columns of 2D vectors representing the vectors along which a 2D normal distribution of covariance matrix $$\Sigma$$ is projected.

This is inspired from the property of 2D normal distributions that the variance of the distribution projected along a vector $$u$$ is $$\sigma^2 = u^T\Sigma u$$ (link). I want to perform tomography, but only care about recovering the 2D covariance matrix.