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- TL;DR Summary
- Reducing an NxN matrix to a 2x2 by application of physical constraints

Gonna preface by saying I never thought linear algebra would be a class I would regret not taking so much... but in short the goal is to reduce an arbitrary symmetric NxN system using a set of auxiliary constraint relationships, e.g. for a 3x3

[tex]

\begin{bmatrix}

V_1\\

V_2\\

V_3\\

\end{bmatrix}

=

\begin{bmatrix}

L_{e11}&L_{e12}&L_{e12}\\

L_{e21}&L_{e22}&L_{e23}\\

L_{e31}&L_{e32}&L_{e33}\\

\end{bmatrix}

\cdot

\begin{bmatrix}

i_1\\

i_2\\

i_3\\

\end{bmatrix}\\

[/tex]

using the following constraints

##V_1=V_2=V_p##

##V_3=V_s##

##i_p=i_1+i_2##

##i_s=i_3##

to end up with an equivalent system with [itex]L_s[/itex], [itex]L_p[/itex], and [itex]M[/itex] in terms of the starting [itex]L_{eij}[/itex] matrix

[tex]

\begin{bmatrix}

V_p\\

V_s\\

\end{bmatrix}

=

\begin{bmatrix}

L_p&M\\

M&L_s\\

\end{bmatrix}

\cdot

\begin{bmatrix}

i_p\\

i_s\\

\end{bmatrix}

[/tex]

For those interested in the context, this is an application specific usage of the method covered in https://onlinelibrary.wiley.com/doi/full/10.1002/eej.23240 but they glossed a bit over some of the key linear math that I don't understand. Eventually I'll be extending this concept to quite large matrices with more complex auxiliary constraints, but for now I'd appreciate some guidance, and some good resources, to get me goin

[tex]

\begin{bmatrix}

V_1\\

V_2\\

V_3\\

\end{bmatrix}

=

\begin{bmatrix}

L_{e11}&L_{e12}&L_{e12}\\

L_{e21}&L_{e22}&L_{e23}\\

L_{e31}&L_{e32}&L_{e33}\\

\end{bmatrix}

\cdot

\begin{bmatrix}

i_1\\

i_2\\

i_3\\

\end{bmatrix}\\

[/tex]

using the following constraints

##V_1=V_2=V_p##

##V_3=V_s##

##i_p=i_1+i_2##

##i_s=i_3##

to end up with an equivalent system with [itex]L_s[/itex], [itex]L_p[/itex], and [itex]M[/itex] in terms of the starting [itex]L_{eij}[/itex] matrix

[tex]

\begin{bmatrix}

V_p\\

V_s\\

\end{bmatrix}

=

\begin{bmatrix}

L_p&M\\

M&L_s\\

\end{bmatrix}

\cdot

\begin{bmatrix}

i_p\\

i_s\\

\end{bmatrix}

[/tex]

For those interested in the context, this is an application specific usage of the method covered in https://onlinelibrary.wiley.com/doi/full/10.1002/eej.23240 but they glossed a bit over some of the key linear math that I don't understand. Eventually I'll be extending this concept to quite large matrices with more complex auxiliary constraints, but for now I'd appreciate some guidance, and some good resources, to get me goin

Last edited: