## QR-Factorization.

### October 28, 2010

While I was studying for a linear algebra exam, I discovered a deep-seeded love for QR-Factorization. I’m not going to explain why this is important, or why we should care about such a factorization; in fact, I have no idea why this is important or why I should care about this. I was directed to this, so I’ll direct you there too.

Anyhow, here’s the game. Given some square matrix (this is not necessarily, but it makes it easier) , we want to decompose into an orthogonal matrix and an upper-triangular matrix .

Just to remind you what this all means, if is an orthogonal matrix, then its columns are orthogonal unit vectors, which means that . A matrix is upper triangular if everything to the left of the diagonal is 0. For example, we could have equal to

which is a nice example of an upper-triangular matrix. Note that, in general, orthogonal matrices are very nice (they have lots of cool properties that I may write a post on one day), and upper-triangular matrices are pretty sweet as well (and I think I may have already written up a post on these…) which is why it’s nice to have any arbitrary square matrix be able to be decomposed as a product of an orthogonal matrix and an upper-triangular one.

## Get to the point already.

Here’s how we do it. Let’s let be our vector space (with inner product ), and is our matrix. Let’s write as

where each is a column vector. Now apply Gram-Schmidt to these, and we obtain an orthonormal set .

Our orthogonal matrix will be equal to

and our upper-triangular matrix will be equal to

which looks messy and complex, but it really isn’t too bad in practice.

## Example!

Let’s do a simple example using the standard real vector space with the inner product as the dot product.

Let’s let be the matrix

Now, we need to apply Gram-Schmidt to the column vectors (which I will write horizontally now, for ease of typing) and . So we obtain and . Note that, in general, you need to divide out by a messy norm, but I was clever here so I didn’t really have to. So now we have our matrices.

We have that is equal to

which is obviously orthogonal, and our is equal to

which is nicely upper-triangular. Now notice that

So sweet. Try this on paper, you’ll really be amazed with how fun it is to do.