It seems that proving the statement given in the question https://math.stackexchange.com/questions/4601532/a-pen-and-paper-proof-for-a-matrix-implication is difficult analytically. I was thinking what sort of different computational approaches we can take to show that the conjecture is true ?

The conjecture is as follows:

Suppose $A = \begin{bmatrix} x & 1\\ y & 0\end{bmatrix}, B = \begin{bmatrix} z & 1\\ w & 0\end{bmatrix}$, for $x,y,z,w \in \Bbb{R}$.

To prove that:

If all the eigen values of $A^2B$ and $AB^2$ are less than one in absolute value $\implies$ $\det(AB+A+I)<0$ and $\det(BA+B+I)<0$ is not possible.

  • 1
    $\begingroup$ There is an interesting article about this, en.wikipedia.org/wiki/Computer-assisted_proof. $\endgroup$ Sep 13, 2023 at 17:18
  • 2
    $\begingroup$ That might be a job for Gröbner bases, but I know almost nothing about them. There is a polynomial condition in the entries of a matrix that is equivalent to all its eigenvalues being smaller than 1 in absolute value, so this conjecture can be expressed in the language of multivariate polynomials and ideals only. $\endgroup$ Sep 13, 2023 at 17:53
  • $\begingroup$ Interesting! I am also not aware of it as well, i was thinking if the question could be framed better or even using the right tags $\endgroup$
    – BAYMAX
    Sep 14, 2023 at 3:18


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.