# $\operatorname{rank}(A) = \operatorname{rank}(B)$, Prove there exist $U, V$ invertible matrices such that: $A = UBV$

$$\DeclareMathOperator{\rank}{rank}\DeclareMathOperator{\Mat}{Mat}$$Given two matrices $$A, B \in \Mat_{m \times n}$$ , as $$\rank(A) = \rank(B)$$.

Prove there exist two invertible matrices: $$U \in \Mat_{m \times m}, V \in \Mat_{n \times n}$$

such that: $$A = UBV$$

My attempt: this question essentially is to prove that multiplying a matrix of the left side is equivalent to be preforming operations on the rows, and multiplying a matrix to the right side is equivalent to be preforming operations on the columns.

I don't know how to prove this - so I tried using Linear maps and to prove that using linear mapps, which was so effective - as this does not "proves" that for every $$A, B$$ with an equal rank, there exist $$U, V$$ so that $$A = UBV$$.

Linear maps make this easier (in my opinion).

As $$\operatorname{rank}(A)=\operatorname{rank}(B)$$, the "column" subspaces $$A(\mathbb{R}^n)$$ and $$B(\mathbb{R}^n)$$ have the same dimension. Let $$U$$ be any linear isomorphism from $$A(\mathbb{R}^n)$$ to $$B(\mathbb{R}^n)$$. Extend $$U$$ to a linear isomorphism of $$\mathbb{R}^m$$.

Now let $$v_1,\ldots,v_k\in\mathbb{R}^n$$ such that $$Av_1,\ldots,Av_k$$ form a basis for $$A(\mathbb{R}^n)$$. Then $$UAv_1,\ldots,UAv_k$$ form a basis for $$B(\mathbb{R}^n)$$. Choose $$w_1,\ldots,w_k\in\mathbb{R}^n$$ such that $$Bw_i=UAv_i$$.

Both $$\ker(A)$$ and $$\ker(B)$$ have dimension $$n-\dim(\operatorname{rank}(A))=n-k$$. Let $$v_{k+1},\ldots,v_n$$ be a basis for $$\ker(A)$$, and $$w_{k+1},\ldots,w_n$$ be a basis for $$\ker(B)$$. Then $$\left\{v_1,\ldots,v_n\right\}$$ and $$\left\{w_1,\ldots,w_n\right\}$$ are bases of $$\mathbb{R}^n$$. Define a linear isomorphism $$V:\mathbb{R}^n\to\mathbb{R}^n$$ on the basis as $$Av_i=w_i$$. Then $$A=U^{-1}BV$$.

If $$r$$ is the common rank, pushing just a little bit more the RREF you get that there are invertibile $$U_{1}, V_{1}$$ such that $$U_{1} A V_{1} = C_{r} = \begin{bmatrix} 1 \\ & 1\\ &&&\ddots\\ &&&&1\\ &&&&&0\\ &&&&&&\ddots\\ &&&&&&&0& \dots\\ \end{bmatrix},$$ where there are $$r$$ ones in $$C_{r}$$, and unnamed entries are zero.

Similarly, there are invertibile $$U_{2}, V_{2}$$ such that $$U_{2} B V_{2} = C_{r}.$$

Now the two expressions are equal, so that...

Hint: prove that any matrix $$A$$ of rank $$r$$ can be transformed to $$A=U\begin{bmatrix}I_r & 0\\0 & 0\end{bmatrix}V$$ by invertible $$U$$, $$V$$. (One way to do it quickly is SVD).