Going through this paper:
http://dx.doi.org/10.1016/S0893-9659(00)00169-5
at the bottom of page 407, the authors seem to compute the determinant of a matrix by expanding down the diagonal.
The authors discuss a matrix
$A = \left[ \begin{array}{ccc} a_{11} a_{12} a_{13} \\ a_{21} a_{22} a_{23} \\ a_{31} a_{32} a_{33} \end{array}\right]$
and call $M_{ij}$ the cofactor of $a_{ij}$.
Then at the bottom of page 407 they write
$\det(A) = a_{11} M_{11} + a_{22} M_{22} + a_{33} M_{33}.$
Is expansion down the diagonal possible? Is it possible in some special cases?
Best Answer
You are not quoting the formula correctly. It says $$ \color{red}{d}[\det(A)] = M_{11}\color{red}{d}a_{11} + M_{22}\color{red}{d}a_{22} + M_{33}\color{red}{d}a_{33},\tag{1} $$ where the $\color{red}{d}$ means derivative. The authors are being sloppy here. They actually consider a matrix function $A(t)=A(0)-tD$, where $A(0)$ is a constant and stable matrix, $D$ is a constant positive diagonal matrix and $t=k^2\ge0$. The $A$ in $(1)$ is actually $A(t)$ and the $d$ means derivative with respect to $t$. Formula $(1)$ is a direct consequence of Jacobi's formula.