The notation for the eigen-decomposition of the
component covariance matrices is as follows
\[
\Sigma_j= \lambda_j^{1/p} \Omega_j \Gamma_j \Omega_j' \qquad j=1, 2, \ldots, k
\]

The dimension of matrices $\Omega_j$ (rotation) and $\Gamma_j$ (shape) is $p\times p$.

$c_{det}=$ scalar, constraint associated with the determinants.

$c_{shw}=$ scalar, constraint inside each group of the shape matrix.

$c_{shb}=$ scalar, constraint among groups of the shape matrix.

Note that if you impose equal volumes $c_{det}=1$. Similarly, if you
impose a spherical shape $c_{shw}=1$.

We also denote with

[

**1**] $\Sigma_B$ the 3D array of size $p\times p \times k$ containing the
empirical covariance matrices of the $k$ groups, before applying the
constraints coming from the 14 parametrizations. In the code $\Sigma_B$
is called $SigmaB$. The $j$-th slice of this 3D array of size $p\times
p$ is denoted with symbol $\hat \Sigma_j$.

[

**2**] $\Omega$ the 3D array of size $p\times p \times k$ containing the
rotation matrices of the $k$ groups. In the code $\Omega$ is called
$OMG$. The $j$-th slice of this 3D array of size $p\times p$ is called
$\hat \Omega_j$.

[

**3**] $\Gamma$ the $p\times k$ matrix containing in column $j$, with
$j=1, 2, \ldots, k$, the diagonal elements of matrix $\Gamma_j$ (shape
matrix of group j). In the code matrix $\Gamma$ is called GAM.

After the application of this routine, the product of the elements of
each column of matrix GAM is equal to 1.

The ratio among the largest (second largest, ...smallest) elements of
each column is not greater than $c_{shb}$ (pa.shb).

The ratio of the elements of each column is not greater than
$c_{shw}$ (pa.shb). All the columns of matrix GAM are equal if the second
letter of modeltype is E. All the columns of matrix GAM are
equal to 1 if the second letter of modeltype is I.

[

**4**] niini the vector of length $k$ containing the number of units
(weights) associated to each group.

[

**5**] $\lambda$ = the vector of length $p$ containing in the $j$-th
position $\lambda_j^{1/p}=|\Sigma_j|^{1/p}$. In the code vector
$\lambda$ is called $lmd$.

The elements of lmd satisfy the constraint pa.cdet in the sense that
$\max(lmd) / \min(lmd) \leq pa.cdet^{(1/p)}$. In other words, the
ratio between the largest and the smallest determinant is not
greater than pa.cdet. All the elements of vector lmd are equal
if modeltype is E** or if $c_{det}=1$ (pa.cdet=1).