QR decomposition: examples

Consider the matrix

This matrix is full column rank. Indeed, the matlab command [Q,R]=qr(A,0) yields a and a :

This shows that is full column rank since is invertible.

The command [Q,R]=qr(A) actually produces the full QR decomposition, with now a orthogonal matrix:

We can see what happens when the input is not full column rank: for example let's consider the matrix

( is not full column rank, as it was constructed so that the last column is a combination of the first and the third.)

The (full) QR decomposition now yields:

We observe that the last triangular element is virtually zero, and the last column is seen to be a linear combination of the first and the third. This shows that the rank of (itself equal to the rank of ) is effectively .