I am studying the fastest-mixing Markov chain in Stephen Boyd & Lieven Vandenberghe's Convex Optimization, but I am stuck trying to understand the following derivation.
How does it calculate the following spectral norm?
$$\| Q P Q \|_2 = \left\| P - \frac 1n 11^\top \right\|_2$$
Can someone put a glance and clear it to me. Will be appreciated.
$$\begin{align*} r &= ||QPQ||\\ &= ||(I -(1/n)\mathbf{1}\mathbf{1}^T)P(I -(1/n)\mathbf{1}\mathbf{1}^T)|| & \text{// definition}\\ &= ||(I -(1/n)\mathbf{1}\mathbf{1}^T)(P -(1/n)(P\mathbf{1})\mathbf{1}^T)||\\ &= ||(I -(1/n)\mathbf{1}\mathbf{1}^T)(P -(1/n)\mathbf{1}\mathbf{1}^T)|| & \text{// } P\mathbf{1} =\mathbf{1} \\ &= ||(P -(1/n)\mathbf{1}\mathbf{1}^TP -(1/n)\mathbf{1}\mathbf{1}^T+(1/n^2)\mathbf{1}\mathbf{1}^T\mathbf{1}\mathbf{1}^T|| & \text{// Distrib}\\ &= ||(P -(1/n)\mathbf{1}(P^T\mathbf{1})^T -(1/n)\mathbf{1}\mathbf{1}^T+(1/n)\mathbf{1}\mathbf{1}^T|| & \text{// } \mathbf{1}^T\mathbf{1} = n\\ &= ||(P -(1/n)\mathbf{1}\mathbf{1}^T -(1/n)\mathbf{1}\mathbf{1}^T+(1/n)\mathbf{1}\mathbf{1}^T|| & \text{// } P=P^T,\quad P\mathbf{1}=\mathbf{1}\\ &= ||(P -(1/n)\mathbf{1}\mathbf{1}^T|| \\ \end{align*}$$