In a report I am reading at work, the error matrix is calculated as the inverse of the hessian matrix and used to calculate the error ellipse angle and axes with a not theoretically correct formula.
They start from a function of 2 variables: $f=f(x, y)$ (I won't use the real function, for the purpose of the question a generic one should be enough) and calculate the hessian matrix:
$$ H = \begin{pmatrix} \partial^2_x f & \partial_x \partial_y f \\\\ \partial_y \partial_x f & \partial^2_y f \end{pmatrix} $$
and the error matrix as its inverse:
$$ E = H^{-1} = \begin{pmatrix} E_{xx} & E_{xy} \\\\ E_{yx} & E_{yy} \\ \end{pmatrix} $$
Written in this way it seems to me that the error associated to the variable $x$ is the one located in position $E[0, 0]$, but if I calculate explicitly the inverse of the hessian:
$$ H^{-1} = \frac{1}{\mid H\mid} \begin{pmatrix} \partial^2_y f & -\partial_x\partial_y f \\\\ -\partial_y\partial_x f & \partial^2_x f \\ \end{pmatrix} $$
the term in position $[0, 0]$ is something related to the $y$ variable and not $x$.
And finally the error matrix is used to calculate the orientation of the error ellipse, $\vartheta$, and the size of the major ($e_1$) and minor ($e_2$) axes with the not theoretically correct formula:
$$ \tan{\vartheta} = \frac{E_{xy}}{E_{xx}} $$
$$ e^2_1 = \frac{E^3_{yy} + E^2_{xy} (2E_{yy} + E_{xx}) }{E^2_{xy} + E^2_{yy} } $$
$$ e^2_2 = \frac{E_{yy}( E_{xx} E_{yy} - E^2_{xy}) }{E^2_{xy} + E^2_{yy} } $$
Now I am confused on what the elements of the error matrix are.
Of course call something $x$ or $y$ is just a convention, but when the error matrix is calculated as inverse of the hessian matrix, its first line is associate to which variable? $x$ or $y$? I.e. is $E_{xx}$ the variance related to the $x$ even if contain term related to the $y$ variable, or it is the $y$ variance?
Did anyone see these equations before and know where I can find any info about them? I couldn't find them anywhere, I always used the eigenvalues and eigenvectors of the matrix never this approximated formula.
I know these are questions to the people who wrote this report, but they are gone leaving just this really confusing document and I don't have anyone else to ask.
