Generalized Diagonal 
I was given the following definition:
  
  For all $\sigma\in S_n$, the product $\prod_\limits{i=1}^n a_{i,\sigma(i)}$ contains one entry from every row and every column, the entries of these matrix is called a "Generalized Diagonal".

What is Generalized Diagonal? is it all the entries of a given $\sigma\in S_n$? what are the uses of it?
 A: I guess it means the sequence $\left(a_{1,\sigma\left(1\right)}, a_{2,\sigma\left(2\right)}, \ldots, a_{n,\sigma\left(n\right)}\right)$ for a given $\sigma \in S_n$. I have never heard of this notion; it is not standard. But it isn't completely useless: For instance, it allows you to say that "the determinant of a lower-triangular matrix is the product of its diagonal entries, because every generalized diagonal other than the main diagonal has at least one zero entry".
A: This resembles the notation used in the Leibniz formula for determinants.

If I define the  signed generalized diagonal product as
$$\operatorname{SGDP}_\sigma = \operatorname{sign}(\sigma) \prod_\limits{i=1}^n a_{i,\sigma(i)}$$
where $\operatorname{sign}(\sigma) = +1$ if $\sigma$ is an even permutation and $\operatorname{sign}(\sigma) = -1$ if $\sigma$ is an odd permutation, then I can make the statement:

The determinant of a matrix is equal to the sum of all its signed generalized diagonal products

Or simply, $\det(A) = \sum_{\sigma \in S_n} \operatorname{SGDP}_\sigma$

As a concrete example, here is the formula for the determinant of a $3 \times 3$ matrix:
$$\det(A)  = a_{1,1}a_{2,2}a_{3,3} - a_{1,2}a_{2,1}a_{3,3}  - a_{1,1}a_{2,3}a_{3,2} + a_{1,2}a_{2,3}a_{3,1} + a_{1,3}a_{2,1}a_{3,2} - a_{1,3}a_{2,2}a_{3,1} $$
