
Let \( M = \left(I_n - \frac{1}{n} 11^T \right) \) be a matrix where \( 1 = (1,1,\dots,1)^T \in \mathbb{R}^n \) and \( I_n \) is the identity matrix of order \( n \). The value of \[ \max_{x \in S} x^T M x \] where \[ S = \{ x \in \mathbb{R}^n \mid x^T x = 1 \} \] is ________.

| P(.) | |
| U = 0 | 0.5 |
| U = 1 | 0.5 |
| P(V = 0| .) | P(V = 1| .) | |
| U = 0 | 0.5 | 0.5 |
| U = 1 | 0.5 | 0.5 |
| P(W = 0| .) | P(W = 1| .) | |
| U = 0 | 1 | 0 |
| U = 1 | 0 | 1 |
| P(Z = 0| .) | P(Z = 1| .) | ||
| V = 0 | W = 0 | 0.5 | 0.5 |
| V = 0 | W = 1 | 1 | 0 |
| V = 1 | W = 0 | 1 | 0 |
| V = 1 | W = 1 | 0.5 | 0.5 |
Consider two distinct positive numbers \( m, n \) with \( m > n \). Let \[ x = n^{\log_n m}, \quad y = m^{\log_m n}. \] The relation between \( x \) and \( y \) is -