Matrix decomposition

Matrix decomposition is the matrix equivalent to algebraic factorization. In this section, we will discuss methods that break a single matrix into the product of two or more smaller matrices.

QR decomposition

QR decomposition is a decomposition that breaks a matrix M into two different matrices, Q and R, such that M is equal to QR. Q is an orthogonal matrix (a matrix in which the inverse of the matrix is equal to its transposition), and R is an upper triangular matrix. Here, we demonstrate QR factorization in R on R's internal trees dataset using the qr command:

> data(trees)
> head(trees)
  Girth Height Volume
1   8.3     70   10.3
2   8.6     65   10.3
3   8.8     63   10.2
4  10.5     72   16.4
5  10.7     81   18.8
6  10.8     83   19.7
> trees.qr <- qr(trees[,c(2:3)])

We recover the Q and R matrices with the qr.Q and qr.R commands, respectively. If we multiply these together, we see that we get our starting dataset as follows:

> Q <- qr.Q(trees.qr...