I am looking to scale a set of conditional eigenvalues calculated for a normalized dataset back to the original basis of the data.
Please bear with me as linear algebra is not my forte! Please feel free to correct me in places where I have gone wrong.
a normalized set of data, $Z$, where;
$Z = \frac{A - E(X)}{\sigma(X)}$
the expectation values and standard deviations, $E(X)$ and $\sigma(X)$ are known while $A$ is an unknown dataset. I am using the normalized dataset since I am comparing variables that are not the same unit.
the associated covariance matrix, $\Sigma(Z)$, where:
$\Sigma_{norm}(Z) = \frac{1}{n-1}Z^{T}Z$
the conditional covariance, $\Sigma_{norm}(X |Y)$, where:
$Z = \left( \begin{array}{c} X \\ \hline Y \end{array} \right) $
$\Sigma_{norm}(Z) = \left( \begin{array}{c|c} J & L^{T} \\ \hline L & K \end{array} \right) $
$\Sigma_{norm}(X |Y) = J - L^{T}K^{-1}L$,
The 'conditional eigenvalues' of the conditional matrix
$\Sigma_{norm}(X |Y) = U_{norm}^{T}D_{norm}U_{norm}$
In the normalized basis, the radii of the hyperellipse are defined by:
$r_{i} = \sqrt{\lambda_{i}}U_{i}$
In my case this gives a 3D Ellipse as follows for a certain Z:
I wish to obtain the same ellipse, scaled to the basis of $A$.
My attempt at a solution:
I notice that calculating the covariance matrix for Z is the same as calculating the correlation matrix. The covariance matrix and correlation matrix are related according to:
$\Sigma_{Norm}(X) = (\text{diag}(\Sigma(X))^{\frac{-1}{2}}\Sigma(X)(\text{diag}(\Sigma(X)))^{\frac{-1}{2}}$
Since $(\text{diag}(\Sigma))^{\frac{-1}{2}}$ is symmetric, I can obtain the unnormalized covariance matrix by using:
$\Sigma(X) = (\text{diag}(\Sigma(X)))^{\frac{1}{2}}\Sigma_{norm}(X)(\text{diag}(\Sigma(X)))^{\frac{1}{2}}$
Ideally then, I need to calculate $\text{diag}(\Sigma(X|Y))^{\frac{1}{2}}$. Is there a way to do this that does not require the calculation of the covariance matrix?
Can anyone offer any advice?
Kindest regards!