We know that in general a projection operator from an n-dimesion vector space V onto an s-dimensional subspace S with basis ${|\alpha_0\rangle ,\ldots, |\alpha_{s-1}\rangle}$ is defined as $P_s = \sum_{i=1}^{s-1} |\alpha_i\rangle\langle\alpha_i|$. From this, we can see that the matrix representation of the projection operator must be a symmetric matrix. Furthermore, we can see that the coefficient of each element must be real for all bases in S (due to the fact that all bases can be written as a linear combination of the standard bases), and 0 for all standard bases in V that are not in S. Thus, all of the elements in the matrix form of P are real and their own complex conjugate. Because P is a symmetric matrix, it is its own transpose. Combining these two facts, we see that P must be its own adjoint.

Ex4 3 / Discussion