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https://github.com/triqs/dft_tools
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56 lines
1.7 KiB
ReStructuredText
56 lines
1.7 KiB
ReStructuredText
.. highlight:: c
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How does it work ?
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###################
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Cofactors
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==========
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For any :math:`n\times n` matrix :math:`A`:
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.. math:: A\,{\rm Cof}(A^T) = {\rm Det}A\, I_n.
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where :math:`\rm{Cof}` means the matrix of the cofactors.
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.. math:: {\rm Cof}(A)_{i,j}
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=(-1)^{i+j}{\rm Det}\begin{pmatrix}
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a_{1,1} & \dots & a_{1,j-1} & a_{1,j+1} & \dots & a_{1,n} \\
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\vdots & & \vdots & \vdots & & \vdots \\
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a_{i-1,1} & \dots & a_{i-1,j-1} & a_{i-1,j+1}& \dots & a_{i-1,n} \\
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a_{i+1,1} & \dots & a_{i+1,j-1} & a_{i+1,j+1}& \dots & a_{i+1,n} \\
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\vdots & & \vdots & \vdots & & \vdots \\
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a_{n,1} & \dots & a_{n,j-1} & a_{n,j+1} & \dots & a_{n,n} \end{pmatrix}.
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Change in the determinant when one adds a line and a column
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============================================================
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: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`:
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.. math:: A'=\begin{pmatrix}
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A & B \\
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C & D \end{pmatrix}.
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Using the previous formula with the cofactors, we get
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.. math:: \frac{{\rm Det}A'}{{\rm Det}A}=C A^{-1} B+D.
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Change in the inverse when one adds a line and a column
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==========================================================
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Using the following variables:
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.. math:: \xi=D-C A^{-1} B, \qquad B'=A^{-1}B, \qquad C'=CA^{-1},
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We get the inverse of the new matrix as:
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.. math:: (A')^{-1}=
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\begin{pmatrix}
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A^{-1}+\xi B'C' & -\xi B'\\
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-\xi C' & \xi
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\end{pmatrix}
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