det_by_minor¶

Determinant Using Expansion by Minors¶

ell¶

We use \(\ell\) to denote the row and column dimension of the square matrix under consideration.

Algorithm¶

This algorithm computes \(|A|\) the determinant of a square matrix \(A \in \B{R}^{\ell \times \ell}\) . In the special case where ell is one, the determinant is just the element of the matrix. Otherwise, the determinant is computed using the formula

\[| A | = + | A(0,0) | - | A(0,1) | + | A(0,2) | \cdots \pm | A(0,\ell-1) |\]

where the last term is + (-) if ell is odd (even) and \(A(i,j) \in \B{R}^{(\ell-1) \times (\ell-1)}\) is the \((i,j)\) minor of A; i.e., the square sub-matrix formed by deleting the i-th row and j-th column of A .

option¶

The only option used by this algorithm is n_arg ; see below:

n_arg¶

This is the number of arguments to the algorithm which is the number of elements in the matrix; i.e., \(\ell^2\) . There is an assert checking that n_arg > 0.

n_other¶

This option is not used except that it may be checked to make sure it is zero.

Implementation¶

cpp_det_by_minor , py_det_by_minor .

Derivative¶

The partial derivative of \(|A|\) with respect to the \((i,j)\)-th element \(A_{i,j}\) is

\[\frac{ \partial |A| }{ \partial A_{i,j} } = (-1)^{i+j} | A(i,j) |\]