P( n by n) be the matrix of ones along the reverse diagonal and zero elsewhere , this is a permutation matrix, hence det(P) = +1 or -1, det() stands for the matrix determinant.

It can be observed that P(n by n) *P(n by n) = Identity (n by n) regardless of n is even or odd.

therefore: det(A - lamda*P) = (+1 or -1) *det( A*P - lambda*Identity(n by n)), for any square matrix A ( n by n).

https://math.stackexchange.com/questions/227345/name-for-diagonals-of-a-matrix

Important or not? As a question it is too general, no definite answer could be given. It depends.

Here is what we find on Google research concerning the main diagonal

,

[Central Connecticut State University

https://chortle.ccsu.edu › vmch13 › vmch13_17

main diagonal

If the matrix is ​​A, then its main diagonal are the elements whose row number and column number are equal, ajj.]

{{The other diagonal of a matrix does not matter and has no name.}}

We assume that this statement is false, but one of the most common mathematical errors.

So a question arises: what is the importance of the LHS diagonal?

One of the important applications of the other diagonal (we call it LHS Diagonal)

[is that it can accurately derive the well-known normal/Gaussian distribution law in a direct and simple way as explained in detail in references 1 and 2.]

Ref

1&2-I.M.Abbas, Do Probability and Statistics Belong to Physics or Mathematics,

Researchgate Mars 2023 and IJISRT review Mars 2023.