Finding Matrices that you can multiply wrong, right
24 days ago
- #determinants
- #matrices
- #linear-algebra
- 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.