Page 250 - A Course in Linear Algebra with Applications

234 Chapter Seven: Orthogonality in Vector Spaces

form an orthonormal basis of Pa(R).

QR-factorization
In addition to being a practical tool for computing or-
thonormal bases, the Gram-Schmidt procedure has important
theoretical implications. For example, it leads to a valuable
way of factorizing an arbitrary real matrix. This is generally
referred to as QR-factorization from the standard notation for
the factors Q and R.

Theorem 7.3.5
Let A be a real m x n real matrix with rank n. Then A can
be written as a product QR where Q is a real m x n matrix
whose columns form an orthonormal set and R is a real nxn
upper triangular matrix with positive entries on its principal
diagonal.

Proof
Let V denote the column space of the matrix A. Then V is
m
a subspace of the Euclidean inner product space R . Since
A has rank n, the n columns X\,... ,X n of A are linearly
independent, and thus form a basis of V. Hence the Gram-
Schmidt process can be applied to this basis to produce an
orthonormal basis of V, say Y±,..., Y n.
Now we see from the way that the Yi in the Gram-Schmidt
procedure are defined that these vectors have the form

f
Y l=b 11X 1
< Y 2 = b 12X 1 + b 22X 2

\ Y n = bi nXi + bi nX2 + • • • + b nnX n

for certain real numbers b^ with ba positive. Solving the
equations for Xi,.. ., X n by back-substitution, we get a linear