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Let $W$ be a subspace of $\mathbb{R}^n$. For $x \in \mathbb{R}^n$, define

$$\rho (x) = \inf_{y \in W} \Vert x - y \Vert _2$$

Let $\{ u_1, \ldots, u_m \}$ be an orthonormal basis of $W$, where $m$ is the dimension of $W$. Extend this to an orthonormal basis $\{ u_1, \ldots, u_m, \ldots, u_n \}$ of all of $\mathbb{R}^n$.

How I can show that

$$\rho (x) = \left[ \mathop {\sum} \limits_{j=m+1}^n \vert \langle x, u_j \rangle \vert^2 \right]^\frac{1}{2}$$

and that it is uniquely attained at

$$y=Px, \qquad P = \mathop {\sum} \limits_{j=1}^m u_j u_j^T$$

As you know, $P$ is called the orthogonal projection of $x$ onto $W$.

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    $\begingroup$ \inf is already a defined operator ;) $\endgroup$ – AlexR Mar 27 '15 at 13:21
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Note that $x = \sum_{i=1}^n \langle u_i, x\rangle u_i$ and by assumption $y = \sum_{i=1}^m \langle u_i, y\rangle u_i$ Thus $$\|x-y\|_2^2 = \sum_{i=1}^n \langle x- y, u_i\rangle^2 = \sum_{i=1}^m \langle x-y, u_i \rangle^2 + \sum_{i=m+1}^n \langle x, u_i\rangle^2 \ge \sum_{i=m+1}^n \langle x, u_i\rangle^2$$ This bound is attained when $\langle x-y, u_i\rangle = 0$ for $1\le i\le m$, i.e. for $y = \sum_{i=1}^m \langle x, u_i\rangle u_i = \sum_{i=1}^m u_iu_i^T x = Px$. This proves $$\rho(x) = \sqrt{\inf_{y\in W} \|y-x\|_2^2} = \sqrt{\|Px-x\|_2^2} = \left(\sum_{i=m+1}^n \langle x, u_i\rangle^2\right)^{\frac12}$$ as claimed. As a side note, this shows that the $\inf$ is actually a $\min$, wich only happens in a finite-dimensional setting.

The uniqueness follows from the definiteness of the inner product. $$\|y-x\|_2^2 = \|Px-x\|_2^2 \Leftrightarrow \langle y-x, u_i\rangle = 0 \quad\forall\ 1\le i\le m\\ \Leftrightarrow \langle y - Px, u_i\rangle = 0\quad\forall\ 1\le i\le m\\ \Leftrightarrow \langle y - Px, y- Px\rangle = 0\\ \Leftrightarrow y = Px$$

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(1) For any $x$, $$ x= \sum_{i=1}^n a_i u_i,\ |x|^2=\sum_{i=1}^n a_i^2 $$

If $ P(x):=\sum_{i=1}^m a_i u_i$ then $$ P(x)\in W$$

If $y\in W$ then $$ y=\sum_{i=1}^m b_i u_i \in W$$ So if $$ |P(x)-x| \geq |y-x| $$ then $$ \sum_{i=m+1}^n a_i^2 \geq \sum_{i=1}^m ( a_i - b_i)^2 + \sum_{i=m+1}^n a_i^2 $$ Note that inequality does not hold, and equality holds when $y=P(x)$. Hence $P(x)$ attains minimum.

(2) Now we will write $P$ as a matrix form : $$ (\sum_{i=1}^m u_i u_i^T - P)(x)=0$$ for all $x$ Hence $P=\sum_{i=1}^m u_i u_i^T$.

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    $\begingroup$ You forgot the uniqueness. $\endgroup$ – AlexR Mar 27 '15 at 13:36

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