Question: I was reading the following
[How do i prove that every open set in $\mathbb{R}^2$ is a union of at most countable open rectangles?](https://math.stackexchange.com/questions/271420/how-do-i-prove-that-every-open-set-in-mathbbr2-is-a-union-of-at-most-count)

The answer by "user4594" basically considers a k-cell with rational endpoints, and I am guessing that when he says that "E would be the union of these cells" the axiom of choice is needed because for each x there may be more than one k-cell with rational endpoints.

Am I correct?

Answer: 
If you write for the row-reducing process the matrix equation using the letters as in user7530's answer
$$ EM = U \qquad \qquad \text{ this alters the eigenvalues of }M$$ 
then a ***correct*** solution which ***preserves the eigenvalues*** were, if you do a ***similaritiy transform***.
Thus if you found some $E$ instead such that
$$ EME^{-1} = U \qquad \qquad \text{ this preserves the eigenvalues of }M \tag 1$$ 
then the eigenvalues are preserved. 

But for this case, $E$ is not easy to find. However I think I remember an iterative process, with which you can find (="approximate") $E$: 

---

Step 0: Let $B\_0 = M$  

Step 1: Then find $E\_0$ (by your set of row-reduction-operations) such that $$ E\_0 B\_0 = U\_0 \qquad \qquad U\_0 \text{ is triangular } \tag 2$$
Step 2: Then compute $ B\_1 = E\_0 B\_0 E\_0^{-1} $

Step 3: Then find $E\_1$ (by your set of row-reduction-operations) such that $$ E\_1 B\_1 = U\_1 \qquad \qquad U\_1 \text{ is triangular }$$
Step 4: and iterate from step 2. 

---

This should approximate $B\_\infty$ to a triangular matrix with the same eigenvalues as $M$. The product $ E = \cdots E\_2 E\_1 E\_0 $ gives then
$$ E M E^{-1} = B \qquad \qquad B \text{ is triangular in the limit with eigenvalues of } M$$

Addendum: If you take care, that the $E\_k$ are rotation-matrices (=orthogonal) you can replace the matrix inversion of $E^{-1}$ by transposition $E^T$ only 

Here is a picture of the iteration-process. All matrices are 4x4; the iterated $B\_k$ matrices are in the left 4x4-column, the $U\_k$ matrices in the second 4x4-columns; the $E\_k$ matrices in the third column and the $E$ matrix as the running partial product in the last column. Ten iterations were needed to approximate $B\_{10}$ to triangular shape with 2 dec digits visible: 

> 
> 
> ```
>     $B_k$             $U_k$                $E_k$                $E$        
> 
> ```
> 
> ![picture](https://i.stack.imgur.com/v6MRQ.png)
> 
> 
>
If a matrix can be row reduced into a triangular form, then you have the eigenvalues already.

$BUT$ every matrix can't be reduced into a triangular form. 

They can be factorized into two matrices, a lower triangular and upper triangular (by LU factorization, you might be aware of it already). But, the diagonals of those matrices won't give you eigenvalues.

$Ax = \lambda x$ OR characteristic equation det|A -$\lambda$I| = 0 must be solved to find the eigenvalues

Also, there are other methods to find the complete eigensystem for a matrix.