A matrix $a\in GL_{n}(F)$ is said to be monomial if each row and column has exactly one non-zero entry. Let $N$ denote the set of all monomial matrices.
I want to prove that following are equivalent
- $A\in N$
- there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=DP$
- there exist a non singular $D$ (diagonal matrix ) and a permutation matrix $P$ such that $A=PD$
I have written a small proof. Is it correct ?
Let's say $A\in N$. We know that "If $P$ is a permutation matrix then multiplication by $P$ on right rearranges the columns." So by multiplying suitable permutation matrix say $Q$ on right we get a diagonal matrix (which is obviously non-singular as $A\in N$). Precisely $AQ=D$. Multiplying by $Q^{T}$ on both sides we get that $A=DP$ for some permutation matrix $P$ and diagonal matrix $D$.
Now assume $A=DP$. Then $$DP = PP^{T}DP=PD^{'}$$ where $D^{'}=P^{T}DP$ is again a diagonal matrix, so $A=PD^{'}$.
Now assume $A=PD$. We want to show that $A\in N$, but it is clear as multiplication on left by $P$ will just swap the rows of $D$.
Your proof is alright. You are using the fact about what happens to a matrix when it is multiplied by a permutation matrix. You can change the view point and use the description of what happens to a matrix when it is multiplied by a diagonal matrix.
If $X, D$ are square matrices of the same size with $D$ diagonal, $XD$ is the matrix obtained from $X$ with its $j$-th column a scalar multiple of corresponding column of $X$ (the scalar being the $j$-th entry of the diagonal matrix).
Analogous statement is true for $DX$. So all monomial matrices are obtained from permutation matrices by multiplying with diagonal matrices.