Problem:
Given the equivalence relation, defined on $\mathbb{R}^{n\times n}$:
$A \sim B$ $\Leftrightarrow$ $\exists S,T \in GL_n(\mathbb{R}): S B T$
Show that there are n+1 equivalence classes of "$\sim$" on $\mathbb{R}^{n\times n}$. Specifically $[0]_\sim$ and $[M_k]_\sim$ for $1 \leq k \leq n$, where $M_k$ is a block matrix of the form:
$$M_k := \begin{pmatrix} I_k & 0 \\ 0 & 0 \end{pmatrix} \in \mathbb{R}^{n \times n} $$
Questions:
Hello ! I currently have the problem of proving that the equivalence classes are exactly as given on the problem itself. I got hints that i should think about using the Gauss-Algorithm as well as elementary row and column operations to figure it out. Yet right now I do not know where to even start. I would require some help understanding this problem in general.
Thanks for the help in advance.
Let $(e_1,\cdots,e_n)$ be the canonical basis of ${\mathbb R}^n$. Let $A$ be a matrix and $k = \text{rank}(A)$. Let $(u_{k+1},\cdots,u_n)$ be a basis of $\ker A$. Complete this basis in a basis $(u_1,\cdots,u_n)$ of ${\mathbb R}^n$. Let $T$ be the matrix of the linear map such that $T(u_i) = e_i$. Complete the basis $(v_1, \cdots, v_k) = (Au_1,\cdots, A u_k)$ of $\text{Im}(A)$ in a basis $(v_{1},\cdots, v_n)$. Let $S$ be the matrix of the linear map such that $S(e_i) = v_i$. Then one has $A = S M_k T$, hence $A\sim M_k$.