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I was studying about matrices when I cam across the following theorem:

An $m×n$ matrix $B$ is column equivalent to an $m×n$ matrix $A$ if and only if $B=AQ$, where $Q$ is a non-singular matrices of order $n$.

My proof for the above theorem, goes as follows:

If we consider two column equivalent matrix $A$ and $B$, such that $B$ is obtained by a series of elementary column operations on $B$. Then ,we can say, $B=AE_1^tE_2^t...E_k^t$(since, we know that $C_{ij}(A)=A(E_{ij})^t$, $C_{i}(c)(A)=A(E_{i}(c))^t$,$C_{ij}(c)(A)=A(E_{ij}(c))^t$, where $E_{ij}$, etc are elementary matrices). Here, $E_1,E_2,...,E_k$ are elementary row matrices. Now, we know, that, all elementary matrices are non-singular and hence $Q=E_1^t...E_k^t$ is a non-singular matrix. Hence, we can say, $B=AQ$.

Is the above proof correct?However, I don't understand how to prove the converse part of the above proof ,i.e if $B=AQ$, where $Q$ is a non-singular matrix, then, $B$ is column equivalent to $A$? I am trying to use the fact that : $A$ is a non-singular iff $A$ can be expressed as a product of elementary matrices i.e $A=E_1E_2...E_k$, where $E_1,E_2,...,E_k$ are elementary matrices. However, I am not able to find a breakthrough?

Arthur
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  • How do you define column equivalence? Usually the theorem you gave is the definition, so there is nothing to prove. – V.S.e.H. Dec 31 '22 at 14:34
  • @V.S.e.H. I think "column equivalent" usually refers to the fact : if $A$ and $B$ are column equivalent matrices if $A$ can be obtained from $B$ by elementary column operations or vice-versa. – Arthur Dec 31 '22 at 14:55
  • So if $A\sim_{col} B$, then there exists $Q$ nonsingular s.t. $A=BQ$, by definition. Conversely, if $A=BQ$ for $Q$ nonsingular, then $A\sim_{col} B$, by definition. So I guess the only thing you have to prove is that any nonsingular $Q$ can be factored as a finite product of elementary matrices, so you're essentially done. – V.S.e.H. Dec 31 '22 at 15:43
  • @V.S.e.H. Can you please explain what you mean by the symbol $A\sim_{col} B$ – Arthur Dec 31 '22 at 17:34
  • Ah sorry, just means column equivalent :) – V.S.e.H. Dec 31 '22 at 19:24

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rest of the proof is :
if $Q$ is non-singular (invertible), it is column equivalent to identity matrix $I_n$. so by finite elementary column(or row) operations (scale,swap,pivot) on $Q$ you'll reach $I_n$: $$ Q(E_1E_2\dots E_k) =I_n $$ inverse of every elementary operation is an elementary operation itself. $$ Q = (E_1E_2\dots E_k)^{-1} = E_k^{-1} \dots E_2^{-1} E_1^{-1} = \overline{E}_1 \overline{E}_2 \dots \overline{E}_k $$ so $Q$ is product of elementary matrices. your other statements are correct so it completed the proof. (proof of above statement is constructive proof of Gaussian elimination for solving system of equations)

Abolfazl Chaman motlagh
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  • If $Q$ is non-singular (invertible), it is column equivalent to identity matrix $I_n$: is this fact well-known? I am asking this because I knew the following theorem: $Q$ is non-singular (invertible),if and only if it is row equivalent to identity matrix $I_n$. Is this true also for column operations? – Arthur Dec 31 '22 at 17:33
  • row operations and column operations are similar, every fact and theorem to this level that has been proven for rows are also true for columns (square matrices). and no it's not a fact. $A$ is invertible iff $AX=0$ has only trivial solution. and you can use this to proof this theorem. see this for example Link – Abolfazl Chaman motlagh Dec 31 '22 at 18:43
  • Ok, so every fact that's proven for row operations are also essentially true for column operations , correct? This is what you mean, right? Also, I am confused by the word "square matrices" ? Did you mean that "every fact and theorem to this level that has been proven for rows are also true for columns $only in$ case of square matrices"? If so, then it doesn't make any sense...I don't get it? – Arthur Jan 01 '23 at 04:08
  • elementary row operations for rows and columns are similar. for example if you want scale a row, you multiply elementary matrix $E$ from left which is identity except in $k$-th diag element is $s$ not 1. if you do this multiplication from right, you get scaling of $k$-th column. so basically you can do all this elementary operations for column instead of row, hence all theorem using elementary operations and row echelon matrix is valid for column. even if matrix is not square things like row rank and column rank are equal. – Abolfazl Chaman motlagh Jan 01 '23 at 09:32
  • but trivial things differ in non-square matrix, like they are linear operations between different(non isomorph) spaces (different dimensions). – Abolfazl Chaman motlagh Jan 01 '23 at 09:32