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Thread: 2 norm of a projector

  1. #1
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    2 norm of a projector

    Let $\displaystyle P \in \textbf{C}^{m\times m}$ be a projector. We prove that $\displaystyle \left\| P \right\| _{2}\geq 1$ , with equality if and only if P is an orthogonal projector.

    I suppose we could use the formula $\displaystyle \left\| P \right\| _{2}= max_{\left\| x \right\| _ {2} =1} \left\| Px \right\| _{2}$

    and use the fact that $\displaystyle P^{2}=P$

    But I am not sure how to proceed.
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  2. #2
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    Quote Originally Posted by math8 View Post
    Let $\displaystyle P \in \textbf{C}^{m\times m}$ be a projector. We prove that $\displaystyle \left\| P \right\| _{2}\geq 1$ , with equality if and only if P is an orthogonal projector.

    I suppose we could use the formula $\displaystyle \left\| P \right\| _{2}= max_{\left\| x \right\| _ {2} =1} \left\| Px \right\| _{2}$

    and use the fact that $\displaystyle P^{2}=P$

    But I am not sure how to proceed.
    Geometrically, there is a good way to see this. Translating it into algebra is not as easy.

    For any vector $\displaystyle x\in\mathbb{C}^m$, you can write $\displaystyle x = Px + (x-Px)$. This is the unique expression for x as the sum of a vector in the range of P (namely Px) and a vector in the kernel of P ( namely xPx). The projection P is orthogonal if and only if its range and kernel are orthogonal subspaces of $\displaystyle \mathbb{C}^m$.

    If P is orthogonal then you can apply Pythagoras' theorem to deduce that $\displaystyle \|x\|^2 = \|Px\|^2 + \|x-Px\|^2 \geqslant \|Px\|^2$. So $\displaystyle \|Px\|\leqslant\|x\|$ (for all x), which tells you that $\displaystyle \|P\|=1$.

    If P is not orthogonal then you want to find a vector x such that $\displaystyle \|Px\|>\|x\|$. The way to do that is to take x to be a unit vector in the orthogonal complement of the kernel of P. That's the hard part of the problem, and I don't have time now to explain how the argument goes.
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    For the first part, can we say that since P is a projector (assuming $\displaystyle P \neq 0$ ), there is a vector $\displaystyle v \neq 0$ such that $\displaystyle Pv=v$.

    Now take $\displaystyle x = \frac{v}{\left\|v \right \|}$, so $\displaystyle \left\|Px \right \|_{2} = 1$ and using the definition of $\displaystyle \left\|P \right \|_{2} $, we get the inequality ?
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