dft_tools/doc/reference/determinant_manipulation/behind.rst

.. highlight:: c

How does it work ?
###################

Cofactors
==========

For any :math:`n\times n` matrix :math:`A`:

.. math:: A\,{\rm Cof}(A^T) = {\rm Det}A\, I_n.

where :math:`\rm{Cof}` means the matrix of the cofactors. 

.. math::    {\rm Cof}(A)_{i,j}
             =(-1)^{i+j}{\rm Det}\begin{pmatrix}
             a_{1,1}   & \dots & a_{1,j-1}   & a_{1,j+1}  & \dots & a_{1,n}   \\
             \vdots    &       & \vdots      & \vdots     &       & \vdots    \\
             a_{i-1,1} & \dots & a_{i-1,j-1} & a_{i-1,j+1}& \dots & a_{i-1,n} \\
             a_{i+1,1} & \dots & a_{i+1,j-1} & a_{i+1,j+1}& \dots & a_{i+1,n} \\
             \vdots    &       & \vdots      & \vdots     &       & \vdots    \\
             a_{n,1}   & \dots & a_{n,j-1}   & a_{n,j+1}  & \dots & a_{n,n}   \end{pmatrix}.

    
Change in the determinant when one adds a line and a column
============================================================

:math:`A` is an inversible :math:`n\times n` matrix. :math:`A'` is a :math:`(n+1)\times (n+1)` matrix obtained by adding a line and a column to :math:`A`:

.. math:: A'=\begin{pmatrix} 
          A & B \\ 
          C & D \end{pmatrix}.

Using the previous formula with the cofactors, we get

.. math::  \frac{{\rm Det}A'}{{\rm Det}A}=C A^{-1} B+D. 


Change in the inverse when one adds a line and a column
==========================================================

Using the following variables:

.. math:: \xi=D-C  A^{-1} B, \qquad B'=A^{-1}B, \qquad C'=CA^{-1}, 

We get the inverse of the new matrix as:

.. math:: (A')^{-1}=
          \begin{pmatrix}
            A^{-1}+\xi B'C' & -\xi B'\\
            -\xi C'         &  \xi
          \end{pmatrix}