Subspaces' distance = 1 The distance between two subspaces is defined as $\|P_1 - P_2\|_2$, where $P_i$ is the orthogonal projector onto each subspace. 


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*What does it mean when the distance is 1?

*Why does the distance between eigenspaces often (though not always) turn out to be 1?


I became curious about this while doing homework, although the homework doesn't ask these questions. From my experiments with various (eigen)subspaces, it appears that if the subspaces are orthogonal, then the distance will be 1. There should be a very simple explanation for that, although I don't have time to work it out right now (something to do with $\sin\theta = 1$ ?). Orthogonality is not a necessary condition, however. For example, subspaces with the following bases have a distance of 1.
$$  \begin{align*} V_1 &= \left[ \begin{array}{rr} -1 & -1 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array} \right] \\
V_2 &= \left[\begin{array}{r} -2 \\ 4 \\ 1 \\ 2\end{array}\right] \end{align*} $$
My math background is limited and my linear algebra is rusty. This question should be trivial for someone with an intuitive grasp of the matrix norm, which I lack (but want to acquire). Thanks.
 A: 
if the subspaces are orthogonal, then the distance will be 1. 

This is true. In fact, a more general statement is true: if one of subspaces contains a vector orthogonal to the other subspace, then the distance is $1$. Indeed, if $x\in V_1\cap V_2^\perp$, then $P_1x=x$ and $P_2x=0$, hence $\|(P_1-P_2)x\|=\|x\|$. 
In particular, if $V_1$ and $V_2$ have different dimensions, then the distance is $1$, because at least one of two intersections $V_1\cap V_2^\perp$ and $V_1^\perp\cap V_2 $ will be nonempty for dimension-counting reasons. 
Eigenspaces of a symmetric matrix are orthogonal, so the distance between them is $1$
The converse is also true: if the distance is $1$, then one of subspaces contains a vector orthogonal to the other. This is more transparent in another, equivalent, description of the distance: it is the Hausdorff distance between $V_1\cap B$ and $V_2\cap B$, where $B$ is the closed unit ball.
The lecture notes by 
Ming-Hsuan Yang ("EECS 275 Matrix Computation") have more relevant facts and applications.
