SubsectionSage and the QR decomposition
Sage has a built-in command to find the QR decomposistion of a matrix.
Essentially, it does the Gram-Schmidt algorithm under the hood.
If you check the documentation, you will see that the matrix has to be defined
over a special type of ring, so use QQbar.
This matrix should be an orthogonal matrix. Let's check that.
That is machine language for “I am pretty sure that's the identity.”
Those question marks are for machine precision representation of exact numbers.