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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.

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    $\begingroup$ There is an interesting article about this, en.wikipedia.org/wiki/Computer-assisted_proof. $\endgroup$ Sep 13, 2023 at 17:18
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    $\begingroup$ 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

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