Question: Let be a normal operator on a finite-dimensional complex inner product space . Consider spectral decomposition and prove that is invertible if and only if for .
Any pointers would be helpful - I am unable to begin currently.
Question: Let be a normal operator on a finite-dimensional complex inner product space . Consider spectral decomposition and prove that is invertible if and only if for .
Any pointers would be helpful - I am unable to begin currently.
Well, this is ALWAYS true: any operator (of a finite-dimensional vector space ) over any field is invertible iff all its eigenvalues are different from zero, and the proof is painfully simple: is NOT invertible iff is an eigenvaue of .