Method of orthogonalization that preserves invertibility Is there a method of orthogonalization such that, given an invertible matrix $A$ with entries in the real numbers, applying the method and then inverting the result is the same thing as applying the method to $A^{-1}$?
For instance, the Gram-Schmidt process does not have this property; the matrix $\begin{pmatrix} 1 & 1 \\ 1 & 0 \\ \end{pmatrix}$ is a counterexample.
 A: Row operations on a matrix changes its inverse by way of column operations. So to have matching actions on matrix and inverse, the actions necessarily have to be both row and column actions.
But it is even slightly worse than that. If you decide that it is ok to work on rows one way and columns the other, the order must also be reversed. For rows, it is first row, second, etc that orthogonalize the rows below. For the reverse direction on columns it is the last column, second to last, etc. (Though if you care it is possible to make the forward equivalent to the backward, by using orthogonal rotations on pairs of rows, first and last, making them orthogonal before using them to orthogonalize the other rows. It is basically a two row SVD making them orthogonal so that their actions on the other rows are independent thus order invariant.)
$\pmatrix{1 & 1 \\ 1 & 0 }$ has inverse $\pmatrix{0 & 1 \\ 1 & -1 }$ and here is the regular Gram-Schmidt, and the column reverse order Gram-Schmidt:
$$\pmatrix{1 & 1 \\ 1 & 0 } \overset{\text{row & forward}}{\rightarrow } \pmatrix{1 & 1 \\ 0.5 & -0.5 \\ } $$
$$\pmatrix{0 & 1 \\ 1 & -1 } \overset{\text{col & reverse}}{\rightarrow } \pmatrix{ 0.5 & 1 \\ 0.5 & -1 \\ } $$
You can see that the two are inverses of each other, and remain inverses of each other in the results.
