Short Version:
If some can solve the easier to read form as follows, I would be thankful.
\begin{equation} B = \frac{1 - d^{T}Bd}{ K_{1} } A \end{equation}
\begin{equation} B^{T}d = \frac{1 - d^{T}Bd}{K_{1}} ( A^{T}c - K_{2}a ) \end{equation}
Where, B and A are $nxn$ matrices, d, c and a are $nx1$ matrices and $K_{1}$ and $K_{2}$ are constants.
Everything except B is known, and B is to be calculated.
Long Version:
I have an ellipsoid in n dimension and I am working out an equation to calculate the ellipsoid at n-1 dimension when a plane with a normal along the $x_{1}$ axis cuts it.
Intersection of the following two equations:
\begin{equation} (p-c)^{T} * A * ( p-c ) = 1 \end{equation} \begin{equation} p_{1} = a \end{equation}
Where p is a point in the configuration space of $n$ dimensions. $c$ is the center of the ellipsoid of which $A$ is the semi-definite matrix.
Opening the equation (1) in the present known coordinate from equation (2). $$ \begin{bmatrix}{} a-c_{1}, q-\hat{c} \\ \end{bmatrix}^{T} * \begin{bmatrix}{} a_{11}, \hat{a_{1}}^{T} \\ \hat{a_{1}}^{T}, \hat{A} \end{bmatrix} * \begin{bmatrix}{} a-c_{1} \\ q-\hat{c} \end{bmatrix} = 1 $$
Where q is a point in the configuration space of $n-1$ dimensions. $c_{1}$ is the center of the ellipsoid along the $x_{1}$ axis. $\hat{c}$ is the list of center points of the ellipsoid excluding the center at the $x_{1}$ axis. $a_{11}$ is the corner most point of A, $\hat{a_{1}}$ is the list of points along the first row of A, excluding the first point.
After manipulation, I brought equation (3) to the following.
\begin{equation} q^{T}\hat{A}q - 2\hat{c}^{T}\hat{A}q + 2(a-c_{1})\hat{a_{1}}^{T}q = 1 - a_{11}(a-c_{1})^{2} - \hat{c}^{T}\hat{A}\hat{c} + 2(a-c_{1})\hat{a}_{1}^{T}\hat{c} \end{equation}
Now, we wanted an the resulting ellipsoid formed by the Intersection, for which the equation was: \begin{equation} (q-d)^{T} * B * ( q-d ) = 1 \end{equation} \begin{equation} q^{T}Bq - 2d^{T}Bq = 1 - d^{T}Bd \end{equation}
Now, the equations (4) and (6) are for the same ellipsoid, so: \begin{equation} B = \frac{1 - d^{T}Bd}{1 - a_{11}(a-c_{1})^{2} - \hat{c}^{T}\hat{A}\hat{c} + 2(a-c_{1})\hat{a}_{1}^{T}\hat{c}} \hat{A} \end{equation}
\begin{equation} B^{T}d = \frac{1 - d^{T}Bd}{1 - a_{11}(a-c_{1})^{2} - \hat{c}^{T}\hat{A}\hat{c} + 2(a-c_{1})\hat{a}_{1}^{T}\hat{c}} ( \hat{A}^{T}\hat{c} - (a-c_{1})\hat{a}_{1} ) \end{equation}
Everything except B is known, and B is to be calculated.