In the following, let i be an N-dimensional column vector with 1s. For some N-dimensional column vector y, the average value of the components in y is given by

(1/N) i' y.

Let I be the identity matrix with dimension N × N. Consider the matrix

M

^{0}= I − (1/N) i i'

The matrix M

^{0}is idempotent. That the matrix is idempotent means that M

^{0}M

^{0}= M

^{0}

d) Show that

1. M

^{0}is symmetric.

2. M

^{0}is idempotent.

- Ben