I'm a second year Computer science student and in my one mathematical statistics module my professor mentioned the idea of a Conditional inverse of a matrix and a Generalised inverse. He listed the following properties of these inverses respectively.
- Generalised(His definition):
- $A^{g}A$ & $AA^{g}$ are symmetric.
- $AA^{g}A$ = $A$
- $A^{g}AA^{g} = A^{g}$
- $A^{g}$ is unique for given $A$
- $A^{g}$ <=> Moore-Penrose
- Conditional:
- $AA^{c}A$ = $A$
- $A^{c}$ is not unique for a given A
- $A^{g} => A^{c}$, but $A^{c}$ does not imply $A^{g}$
Now from other courses in my degree I know about the Moore-Penrose Inverse my understanding of the Moore-Penrose inverse is that it has these properties.
- Is unique
- $AA^{†}A = A$
- $A^{†}AA^{†} = A^{†}$
- $(A^{†}A)^{T} = A^{†}A$ i.e. symmetric
- $(AA^{†})^{T} = AA^{†}$ i.e. symmetric
And from other sources online I find these definitions/properties of the so called Generalised inverse.
Generalised (from other sources)
- AGA = A
- Not unique
- GAG = G
- $A^{†}$ => G but G does not => $A^{†}$
You might think I just misheard my prof because his definition of the generalised inverse sounds a lot like the actual Moore-Penrose definition and vice-versa and that his conditional inverse sounds like the actual Generalised inverse but I can without a doubt say no these are the definitions that he goes by and tries to teach us. As he mentions these definitions in his notes and repeatedly in lectures. We only follow his notes and there are no references to any sources in his notes.
My question boils down to this. Is the Conditional inverse a well defined inverse(That specifically goes by the name conditional inverse and the symbol $A^{c}$)? Is it used commonly with respect to the other types of inverses? Is his definition correct for the generalised inverse?
I'm a firm believer of not just trusting things blindly,question everything, and would really appreciate the mathematics community input on this.