Prove these properties of the pseudo inverse:
1) $(AA^*)^{\dagger}={A^{\dagger}}^*A^{\dagger}$;
2) $A^{\dagger}=A^*(AA^*)^{\dagger}$.
I'm quite sure I need to use the four properties of the pseudo inverse, but I'm not exactly sure how. Moreover, I also know that in general we cannot expect $A^{\dagger} A=I$.
Here are the four properties that the pseudoinverse must satisfy as listed in my text:
1) $A X A=A$
2) $X A X = X$
3) $(AX)^*=AX$
4) $(X A)^* = X A$
Please Help. Thanks.
We know that $*$ of $\dagger$ is equal to $\dagger$ of $*$. Note that a simple manipulation of the pseudoinverse properties gives $$ A=AA^\dagger A=A(A^\dagger A)^*=AA^*A^{\dagger *}, $$ $$ A^*=(AA^*A^{\dagger *})^*=A^\dagger AA^*. $$ $$ A^\dagger=A^\dagger A A^\dagger=(A^\dagger A)^* A^\dagger=A^*A^{\dagger *}A^\dagger, $$ $$ A^{\dagger *}=(A^*A^{\dagger *}A^\dagger)^*=A^{\dagger *}A^\dagger A, $$
You need to verify the four properties, which can be done easily with the above identities: $$ AA^*(A^{\dagger *}A^\dagger)AA^*=(AA^*A^{\dagger*})(A^\dagger AA^*)=AA^*, $$ $$ (A^{\dagger *}A^\dagger)AA^*(A^{\dagger *}A^\dagger)= (A^{\dagger *}A^\dagger A)(A^*A^{\dagger *}A^\dagger)=A^{\dagger *}A^{\dagger}. $$ The last two properties are simple. Together, this verifies that $A^{\dagger *}A^\dagger$ is the pseudoinverse of $AA^*$. To prove the second statement, have a look on the third identity above.