Solving vv^T = M

daviddoria

I have a known matrix M and an unknown vector outer product: ie v is 3x1 unknown and M is 3x3 known. Clearly there are 9 equations (each entry in vv^T must equal the corresponding entry in M) - but how do you solve this without manually writing those equations?

Thanks,

Dave

Hurkyl

Sheesh, it's only 9 of them! It won't take long -- and if you do so and then try to solve them, a shorter method should present itself.

You can always try the 2x2 case first, if you want to start smaller....

daviddoria

The idea of not just writing it out and looking for a pattern is that even if I see one, should the same problem arise in the 200x1 vector case I am clearly not going to write it out then - I know there are some fancy operators (ie Kroneker product) that do stuff like this and I was hoping someone would have a nice explanation in a "one sentence" kind of math statement instead of writing a big loop or something like I would do.

Dave

HallsofIvy

There are two problems with solving vv^T= M:

1) The mapping is not one-to-one. Different v may give the same M.

2) The mapping is not onto. Since vv^T is necessarily symmetric, for most M there is NO v such that vv^T= M.

daviddoria

HallsofIvy: you are exactly right. However, in this case I am looking for multiple solutions and I know that M is such that these solutions exist.

This is a problem in computer vision - namely 3d reconstruction from two images. It is known that there are 4 solutions to vv^T = M in this case, so you are supposed to get the 4 solutions and then decide which one is physically realizable.

Dave

Hurkyl

... because you already know a pattern, having discovered it by studying the 3x1 case.

(or even having discovered it from just writing out a few of the equations in the 200x1 case)

daviddoria

Ok so I see the pattern...
v_i ^2 on the diagonal and v_ij in the off diagonal entries. Now the problem remains - how do you solve this using matrix techniques (ie in Matlab)? It is now non-linear so I guess there is no chance of doing this (ie. Linear algebra)?

Dave

Hurkyl

Well, it's a quadratic equation, so it would be unlikely that it could be solved purely by solving linear systems. But matlab lets you take square roots, doesn't it? So all that's left is to work out what the signs are supposed to be. Incidentally, I think (but am not sure) there's another trick you can do to avoid having to do even that much work -- I think you can manage by just doing a couple divisions for each element of v.



As an aside, I wouldn't be surprised if Matlab already has a built-in function that computes what you want (if it exists).