Register to reply

How to extract a subspace

by weetabixharry
Tags: extract, subspace
Share this thread:
Nov22-12, 04:04 PM
P: 108
I have a (3 x N) matrix of column rank 2. If each column is treated as a point in 3-space, then connecting the points draws out some planar shape.

What operation can I apply such that this planar shape is transformed onto the x-y axis, so that the shape is exactly the same, but is now described fully by x-y coordinates in a (2 x N) matrix?

I feel like there should be a (2 x 3) matrix that would do this, but I can't figure out what it should be. (I have a hunch that I'm looking for a mapping that is isometric and conformal... some kind of rotation?). Also, I'd like to be able to generalise to higher dimensions.
Phys.Org News Partner Science news on
Climate change increases risk of crop slowdown in next 20 years
Researcher part of team studying ways to better predict intensity of hurricanes
New molecule puts scientists a step closer to understanding hydrogen storage
Nov23-12, 09:14 AM
P: 108
I think I may have found a solution, but would appreciate any further discussion... since my understanding is rather weak. I basically started thinking about pseudoinverses and figured I wanted a pseudoinverse of something orthonormal, to avoid distorting my shape (?).

Let's call my (3 x N) matrix A. To get an orthonormal basis spanning the (2D) column space, I eigendecompose AAT and take the eigenvectors associated with the 2 largest eigenvalues, denoted by the (3 x 2) matrix E2.

Finally, I left-multiply A by the pseudoinverse of E2:

A2D = E2+A

which seems to give the desired 2D representation.
Nov24-12, 12:39 PM
P: 108
Quote Quote by weetabixharry View Post
A2D = E2+A

which seems to give the desired 2D representation.
Of course, this can be simplified as:[tex]\begin{eqnarray*}
\mathbf{A}_{2D} &=&\mathbf{E}_{2}^{+}\mathbf{A} \\
&=&\left( \mathbf{E}_{2}^{T}\mathbf{E}_{2}\right)^{-1} \mathbf{E}_{2}^{T}\mathbf{A%
} \\
However, I still don't really have an intuitive idea for why this works. Perhaps I should re-ask the question in Linear Algebra.

Register to reply

Related Discussions
Prove: sum of a finite dim. subspace with a subspace is closed Calculus & Beyond Homework 1
Dimension of an intersection between a random subspace and a fixed subspace Set Theory, Logic, Probability, Statistics 4
Extract h General Math 6
Finding a subspace (possibly intersection of subspace?) Calculus & Beyond Homework 7
A subspace has finite codimension n iff it has a complementary subspace of dim nu Calculus & Beyond Homework 3