How to project a n-dimensional point onto a 2-D subspace?

Question

Here is a n-dimensional space: There's a point

$P (p_1,p_2,\dots,p_n)$

And two orthogonal vectors that determines a 2-D plane/subspace D

$v (v_1,v_2, \dots ,v_n)$ $w (w_1,w_2, \dots ,w_n)$

How do I project the point $P$ onto the 2-D subspace D that is determined by vector $v$ and $w$?

$\Pi_D P = \frac{\langle P, w\rangle}{\langle w,w\rangle} w + \frac{\langle P, v\rangle}{\langle v,v\rangle} v$ — martini, Aug 22 '12 at 19:29
$\langle P, w\rangle = \sum_i p_iw_i$, the inner product of $P$ and $w$. — martini, Aug 22 '12 at 19:41
@martini : If I'm not mistaken, what you've written works if $v$ and $w$ are mutually orthogonal, but not otherwise. — Michael Hardy, Aug 22 '12 at 20:55
Three upvotes for "martini"'s comment! Does no one else see the error in it? — Michael Hardy, Aug 22 '12 at 20:56
@Michael, $v$ and $w$ are stated to be orthogonal in the question. — , Aug 22 '12 at 21:16
@Hi271 when $v, w$ are orthogonal, then $M^T M = I$ in Michael's answer, and the formula simplify to martini's comment. — , Aug 22 '12 at 22:23
OK, I missed that point. @JenniferDylan : You won't get $I$ unless $v,w$ are unit vectors, but you get a diagonal matrix. — Michael Hardy, Aug 22 '12 at 23:43

score 11 · Accepted Answer · answered Aug 22 '12 at 20:51

If you know matrices, this will do it: Regard $P$, $v$, and $w$ are column vectors. Let $M$ be the matrix whose two columns are $v$ and $w$. It's an $n\times2$ matrix. (By the way, you shouldn't use capital $N$ and lower-case $n$ as if they were synonymous. Mathematical notation is case-sensitive.) Then $M^TM$ is a $2\times 2$ matrix, which is invertible if the vectors $v$, $w$ are linearly independent. The matrix $M(M^TM)^{-1}M^T$ is and $n\times n$ matrix of rank $2$. The vector $M(M^TM)^{-1}M^T P$ is the projection that you seek.

"Usage note": Once upon a time a highly respected and moderately famous mathematician told me that $M(M^TM)^{-1}M^T$ is the identity matrix. Apparently he was assuming $M$ was a square matrix. I have a bold hypothesis, which I haven't checked empirically: "Pure" mathematicians tacitly assume matrices are square; "applied" mathematicians don't.

How to project a n-dimensional point onto a 2-D subspace?

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