11
votes
Accepted
Is the number of similarity classes of integer matrices with given minimal and characteristic polynomial finite?
No. Consider the matrices
$$A_n = \begin{bmatrix} 0&n \\ 0&0 \end{bmatrix}$$
for $n$ a positive integer. They all have the same characteristic and minimal polynomials (namely, $x^2$), but no ...
- 156k
2
votes
Is the number of similarity classes of integer matrices with given minimal and characteristic polynomial finite?
David's answer fully addresses the question I asked, but after considering his examples I realized that we can resolve the finiteness question in general, so I figured I'd share. In particular, we can ...
- 123
1
vote
Is an almost-solvable linear equation with integer coefficients solvable?
Considering $M$ as a linear operator from $\mathbb R^n$ to $\mathbb R^m$, its range $\text{Ran}(M)$ is a linear subspace. Linear subspaces in finite-dimensional spaces are closed. Thus if $b \notin \...
- 54.2k
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