Page 160 - Matrices theory and applications

Second method. We proceed by induction on n. The statement is clear
if n = 1. Otherwise, we seek an L of the form

L
0
,
L =
l
knowing that
X T 8.3. The QR Factorization 143

M R
M = T .
R m

The matrix L is obtained by Choleski factorization of M ,which
belongs to SPD n−1 .Then X is obtained by solving L X = R. Finally,

2
2
l is a square root of m − X .Since 0 < det M =(l det L ) ,wesee
2
2
that m − X > 0; we thus choose l = m − X . This method
again shows uniqueness.
Remark: Choleski factorization extends to Hermitian positive definite ma-
trices. In that case, L has complex entries, but its diagonal entries are still
real and positive.
8.3 The QR Factorization
In this section k = IR or CC, the real case being a particular case of the
complex one.
Proposition 8.3.1 Let M ∈ GL n (CC) be given. Then there exist a unitary
matrix Q and an upper triangular matrix R, whose diagonal entries are real
positive, such that M = QR. This factorization is unique.
We observe that the condition on the numbers r jj is essential for unique-
¯
ness. In fact, if D is diagonal with |d jj | =1 for every j,then Q := QD is

unitary, R := DR is upper triangular, and M = Q R , which gives an in-

finity of factorizations “QU.” Even in the real case, where Q is orthogonal,
n
there are 2 “QU” factorizations.
Proof
We first prove uniqueness. If (Q 1 ,R 1 )and (Q 2 ,R 2 ) give two factoriza-
−1 −1
tions, then Q = R,with Q := Q 2 Q 1 and R := R 2 R 1 .Since Q is unitary,
we deduce Q = R −1 ,or Q = R −∗ . This shows (recall that the inverse of a
∗
triangular matrix is a triangular matrix of same type) that Q is simultane-
ously upper and lower triangular, and is therefore diagonal. Additionally,
2
its diagonal part is strictly positive. Then Q = Q Q = I n gives Q = I n .
∗
Finally, Q 2 = Q 1 and consequently, R 2 = R 1 .
The existence follows from that of Choleski factorization. If M ∈
GL n (CC), the matrix M M is Hermitian positive definite, hence admits a
∗
Choleski factorization R R,where R is upper triangular with real positive
∗