3 Generalized inverse matrices

3. Generalized inverse matrix 3.1 definition

Generalized inverse
AMXN,XMXN a_{m \times n}, x_{m \times n}, if X meets moore-penrose conditions axa=a xax=x (AX) h=ax (ax) ^h=ax (XA) h=xa (XA) ^h=xa
is called X is a generalized inverse matrix, referred to as the generalized inverse

Pseudo-Inverse A + a^+ if x satisfies all of the above moore-penrose conditions, it is said that X is a pseudo-inverse, or a plus inverse (m-p inverse), recorded as A + a^+, if A is reversible, then a−1=a+ a^{-1} = a^+. ∀anxn∈c,a+ \forall a_{n \times n} \in c,a^+ exists and is unique. Properties
Aa+a=a aa^+a=a a+aa+=a+ a^+a a^+= a^+ (aa+) h=aa+ (aa^+) ^h = aa^+ (a+a) h=a+a (a^+a) ^h = A^+a

Operation of Pseudo-inverse
Set Anxn∈c a_{n \times n} \in C, the pseudo inverse of pseudo-inverse is self, (A +)