Page 162 - Matrices theory and applications

8.4. The Moore–Penrose Generalized Inverse
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The interest of this construction lies also in giving a more complete
statement than Proposition 8.3.1:
Theorem 8.3.1 Let M ∈ M n (CC) be a matrix of rank p.There exists
+
for every l and
Q ∈ U n and an upper triangular matrix R,with r ll ∈ IR
r jk =0 for j> p, such that M = QR.
Remarks:The QR factorization of a singular matrix (i.e., a noninvertible
one) is not unique. There exists, in fact, a QR factorization for rectangular
matrices, in which R is a “quasi-triangular” matrix.
8.4 The Moore–Penrose Generalized Inverse
The resolution of a general linear system Ax = b,where A may be singular
and may even not be square, is a delicate question, whose treatment is
made much simpler by the use of the Moore–Penrose generalized inverse.
We begin with the fundamental theorem.
Theorem 8.4.1 Let A ∈ M n×m (CC) be given. There exists a unique matrix
A ∈ M m×n(CC), called the Moore–Penrose generalized inverse, satisfying
†
the following four properties:
1. AA A = A;
†
†
2. A AA = A ;
†
†
†
3. AA ∈ H n ;
4. A A ∈ H m .
†
Finally, if A has real entries, then so has A .
†
When A ∈ GL n (CC), A coincides with the standard inverse A −1 ,since
†
the latter obviously satisfies the four properties. More generaly, if A is
onto, then property 1 shows that AA = I n ; i.e., A is a right inverse of A.
†
†
†
†
Likewise, if A is one-to-one, then A A = I m ; i.e., A is a left inverse of A.
Proof
We first remark that if X is a generalized inverse of A, that is, it satisfies
∗
∗
these four properties, and if U ∈ U n , V ∈ U m ,then V XU is a generalized
inverse of UAV . Therefore, existence and uniqueness need to be proved
for only a single representative D of the equivalence class of A modulo
unitary multiplications on the right and the left. From Theorem 7.7.1, we
may choose D = diag(s 1 ,... ,s r , 0,... ), where s 1 ,... ,s r are the nonzero
singular values of A.
We are thus concerned only with quasi-diagonal matrices D.Let D be
†
any generalized inverse of D, which we write blockwise as

G H
†
D = .
J K