Finding Matrices that you can multiply wrong, right

2 months ago

The post discusses finding NxN matrices A and B where B is fixed for a given A.
A and B share eigenvectors and their eigenvalues are associated, proving AB = BA (matrices commute).
The determinant of B is calculated, suggesting a method to pick A such that B has small positive integers.
B can be written as a linear combination or as an integer polynomial of A.
If A's characteristic polynomial is an integer polynomial, B is forced to be an integer matrix.
Representing B as 10E⁻¹ simplifies the problem to finding an integer matrix E with determinant dividing 10.
Random search for E with determinant ±1 or ±10, followed by determinant-preserving transformations, is suggested.
Matrices where off-diagonal entries are all positive are called Metzler Matrices, which might help in generating desired matrices.

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