Spectral norm of matrix restricted to subspace $1^\perp$

Question

On Page 6 (of 23) of the following document:

Boyd et. al - Fastest Mixing Markov Chain on a Graph

The spectral norm of a matrix P, restricted to subspace $\mathbf{1}^\perp=\{u \in R^{n} \mid \mathbf{1}^Tu=0\}$, is defined to be,

$\|(I - (1/n)\mathbf{11}^T) P (I - (1/n)\mathbf{11}^T) \|_2 = \|P - (1/n)\mathbf{11}^T \|_2$

where, $(I - (1/n)\mathbf{11}^T)$ gives orthogonal projection on subspace $1^{\perp}$, and $\|.\|_2$ denotes the spectral norm.

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My question is: How is the matrix $(I - (1/n)\mathbf{11}^T) P (I - (1/n)\mathbf{11}^T)$, the restriction of P on the subspace $1^{\perp}$ ? Given that restriction of matrix P, by definition, would be a matrix R such that (see answer of mathstackexchange),

$Rx= \left\{ \begin{array}{ll} Px&\mbox{if $x\in 1^\perp$}\\ 0&\mbox{if $x\in 1$} \end{array} \right.$

Answer 1

Neither your definition nor the paper's define that which is typically referred to as a "restriction". The restriction of a linear map to the subspace $U$ is the subspace $U$, but both your map and the paper's has $\Bbb R^n$ as its domain.

The only term I've heard that describes the kind of map defined in the paper is a compression. There is some ambiguity in the use of this term here: if $V$ is an isometry from $\Bbb R^k$ to the subspace $U \subset \Bbb R^n$, then the map $V^*PV$ might also be referred to as the compression of $P$ to $U$. In contrast, the map considered in the paper is equal to $VV^* P VV^*$. Notably, these maps have the same spectral norm.