Max Determinant

Determinants of special square matrices with elements defined as the Maximum or Minimum of the Indices.
math
linear algebra
determinant
Author

Enrique Pérez Herrero

Published

March 5, 2022

Plot3D of max matrix elements

Identity

The sequence A181983(n) gives the determinants of the square matrices: Mn, where the elements mi,j=max(i,j) and max denotes the maximum function.

This identity appears as follows:

det[max(i,j)]1≤i,j≤n=−n⋅(−1)n=A181983(n)

Proof

A new matrix with the same determinant can be created by subtracting row i from row i+1 starting from the 2nd row. The determinant of this new matrix can then be easily computed using the expansion by minors technique at element m1,n

This can be better illustrated with an example:

We can transform:

M10=(12345678910223456789103334567891044445678910555556789106666667891077777778910888888889109999999991010101010101010101010)

Into:

M10∗=(12345678910100000000011000000001110000000111100000011111000001111110000111111100011111111001111111110)

This proof can be generalized to a very similar type of matrices, resulting in:

det[max(i,j)k]1≤i,j≤n=(−1)n−1⋅nk⋅∏s=1n−1(s+1)k−sk

det[min(i,j)k]1≤i,j≤n=∏s=1n−1(s+1)k−sk

References

[1]
N. J. A. Sloane, “A051125: Table t(n,k) = maxn,k read by antidiagonals (n >= 1, k >= 1).” Available: https://oeis.org/A051125/
[2]
M. Somos, “A181983: A(n) = (-1)^(n+1) * n.” 2012. Available: https://oeis.org/A181983/
[3]
E. Deutsch, “A161124: Number of inversions in all fixed-point-free involutions of 1,2,...,2n.” Jun. 2009. Available: https://oeis.org/A161124/
[4]
N. J. A. Sloane, “A001147: Double factorial of odd numbers: A(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).” Available: https://oeis.org/A001147/