One of the conditions for the spectral decompositions $P_{\lambda} (\lambda \in Sp(A))$ of diagonalizable operator $A$ acting upon a finite dimensional vectorspace is $P_{\lambda} P_{\mu} = \delta_{\lambda, \mu} P_{\mu}$.
What does $\delta_{\lambda, \mu}$ mean? The notes that I am using do not define it, and I am struggling to find it online.
It is a the Kronecker delta, a function of $\lambda$ and $\mu$ --- equal to one when $\lambda=\mu$, and zero otherwise. It is expressing that $P_\mu$ and $P_\nu$ are orthogonal ('perpendicular').