hey, the task is as follows:
Show that if the matrix P is idempotent, meaning that [MATH]P=P^2[/MATH] then [MATH]E-P[/MATH] and [MATH]P+AP-PAP[/MATH] is also idempotent. Note that E is the identity matrix.
I start of by squaring E-P and it simplifies nicely, [MATH](E-P)^2=P^2-2EP+E^2=P-2P+E=E-P[/MATH], my problem is with the second one. I know that [MATH]P^n=P[/MATH], [MATH](P+AP-PAP)[/MATH] factors to [MATH]P(E+A-AP)[/MATH] and now I square it: [MATH]P^2((E+A-AP)^2)=P((E+A-AP)^2)[/MATH], expanding this monster I get: [MATH]P(A^2+2AE+E^2-2A^2P-2AEP+A^2P^2)[/MATH] and simplifying I get [MATH]P(A^2+2A+E-2A^2P-2AP+A^2P)[/MATH] and here I don't see any more simplification so I multiply P into the parenthesis: [MATH]PA^2+2PA+PE+2PA^2P-2PAP+PA^2P[/MATH] and here I get stuck, I am guessing that I have to be careful about the order in which I put my matrices and so I am likely doing things I am not actually supposed to be able to do.
Show that if the matrix P is idempotent, meaning that [MATH]P=P^2[/MATH] then [MATH]E-P[/MATH] and [MATH]P+AP-PAP[/MATH] is also idempotent. Note that E is the identity matrix.
I start of by squaring E-P and it simplifies nicely, [MATH](E-P)^2=P^2-2EP+E^2=P-2P+E=E-P[/MATH], my problem is with the second one. I know that [MATH]P^n=P[/MATH], [MATH](P+AP-PAP)[/MATH] factors to [MATH]P(E+A-AP)[/MATH] and now I square it: [MATH]P^2((E+A-AP)^2)=P((E+A-AP)^2)[/MATH], expanding this monster I get: [MATH]P(A^2+2AE+E^2-2A^2P-2AEP+A^2P^2)[/MATH] and simplifying I get [MATH]P(A^2+2A+E-2A^2P-2AP+A^2P)[/MATH] and here I don't see any more simplification so I multiply P into the parenthesis: [MATH]PA^2+2PA+PE+2PA^2P-2PAP+PA^2P[/MATH] and here I get stuck, I am guessing that I have to be careful about the order in which I put my matrices and so I am likely doing things I am not actually supposed to be able to do.