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 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...