I'm here again with a somewhat vague and hard question our teacher asked us, we have to check and proof that for all matrices $A$ and $B$ that by applying certain simultaneous transformations, we can arrive at the possibilities stated in the theorem (this I guess form an equivalence class under an alternative definition of similarity). We have to only prove it for $3 \times 3$ matrices, Some partners said that they are presumably, indecomposable.

\begin{align*} \begin{array}{l}{\text { Theorem. }^{10} \text { Given any two matrices } A, B \in k^{m \times u}, \text { there exist invertible matrices }} \\ {X \in G L_{m}, Y \in G L_{u} \text { such that } X A Y^{-1} \text { and } X B Y^{-1} \text { have the block-forms of Fig. } 16,} \\ {\text { where the pairs }\left(A_{i}, B_{i}\right) \text { are chosen among the following pairs }(U, V) \text { : }}\end{array} \end{align*} a imague of the forms of XAX^-1 and XBX^-1 that show they are block diagonal

a) $U=\left[\begin{array}{c}{1_{n}} \\ {0}\end{array}\right] \quad V=\left[\begin{array}{c}{0} \\ {1_{n}}\end{array}\right] \in k^{(n+1) \times n} \qquad n \geqslant 0$

b) $U=1_{n} \qquad V=J_{n}^{T} \quad n \geqslant 1$

c) $U=s \mathbb{1}_{n}+J_{n} \quad V=1_{n} \quad n \geqslant 1, s \in k$

d) $U=\left[\begin{array}{ll}{1_{n} 0 ]} & {V=\left[0 ~ 1_{n}\right] \in k^{n \times(n+1)} \quad n \geqslant 0}\end{array}\right.$

EDIT: I'll guess $1_n$ is the identity and that $J_{n}$ is the shift matrix of size n.

I'm not really asking for the proofs but at least a concrete version of the possibilities, the ones that he told us were only but confusing, for example, he stated $A$ could always be brought to the identity, (that would be absurd because then all matrices would have det $ \not = 0$) when he probably meant an identity of size $ n \times n $ where $n \leq 3$. I also ask for help because the book isn't at our math level, it's from the Encyclopaedia of Mathematical Sciences 73 in a segment by Gabriel and Roiter, and this is a freshman level course. so any hints would also be appreciated.

  • $\begingroup$ What are $1_n$, $\mathbb{1}_n,J_n?$ Also, in part d), the expression for $V$ looks wrong. $\endgroup$ – saulspatz Jul 8 at 15:25
  • $\begingroup$ Thanks i have corrected it now, and added more details $\endgroup$ – Alejandro Quinche Jul 8 at 15:31

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