Linear Algebra, Theory And Applications, 2012a

130 SOME FACTORIZATIONS
It is customary to write this more simply as
A = PLU
where L is an upper triangular matrix having all ones on the diagonal and P is a permutation
matrix consisting of P 1 ···P n−1 as described above. This proves the following theorem.
Theorem 5.6.1 Let A be any invertible n × n matrix. Then there exists a permutation
matrix P and a lower triangular matrix L having all ones on the main diagonal and an
upper triangular matrix U such that
5.7 The QR Factorization
A = PLU
As pointed out above, the LU factorization is not a mathematically respectable thing because
it does not always exist. There is another factorization which does always exist.
Much more can be said about it than I will say here. At this time, I will only deal with real
matrices and so the inner product will be the usual real dot product.
Definition 5.7.1 An n × n real matrix Q is called an orthogonal matrix if
QQ T = Q T Q = I.
Thus an orthogonal matrix is one whose inverse is equal to its transpose.
Thus
First note that if a matrix is orthogonal this says
∑
Q T ijQ jk = ∑ Q ji Q jk = δ ik
j
j
⎛ ⎞
|Qx| 2 = ∑ ⎝ ∑ 2
Q ij x j
⎠ = ∑ ∑ ∑
Q is x s Q ir x r
i j
i r s
= ∑ ∑ ∑
Q is Q ir x s x r = ∑ ∑ ∑
Q is Q ir x s x r
i r s
r s i
= ∑ ∑
δ sr x s x r = ∑ x 2 r = |x| 2
r s
r
This shows that orthogonal transformations preserve distances. You can show that if you
have a matrix which does preserve distances, then it must be orthogonal also.
Example 5.7.2 One of the most important examples of an orthogonal matrix is the so
called Householder matrix. You have v a unit vector and you form the matrix
I − 2vv T
This is an orthogonal matrix which is also symmetric. To see this, you use the rules of
matrix operations.
(
I − 2vv
T ) T
= I T − ( 2vv T ) T
= I − 2vv T