How to find the equation of a 2D plane that best fits a set of vectors defined in n dimensions?

I have several vectors defined in 6 dimensions that should all lie on the surface of a 2D plane.

I know that the plane can be readily defined using any two vectors in the plane that are not colinear, these two vectors should span the subspace defined by the plane. However, the vectors all have some amount of noise (as they are determined through an experimental procedure). Therefore I'm looking to find the best fitting plane, that best describes all of the vectors, rather than just picking any two.

I think that the plane should either be defined by a vector equation, or by a system of linear equations. I think the latter system should consist of 4 equations in order to entirely constraint the subspace into 2 dimensions (which should be the case, since it is a 2D plane we are finding).

I'm struggling to do this. Does anyone have any ideas? Thanks for your help!

[Swastik Sarkar]

Hi @Rahul,

For determining the best plane of fit for vectors, mean-centering the data followed by applying singular value decomposition (SVD) can work. This approach identifies the plane that optimally fits the given vectors. The process is detailed in the following MATLAB Answer:

https://www.mathworks.com/matlabcentral/answers/496215-svd-and-basis-of-a-plane

[Rahul]

Excellent, thank you! This was perfect.

Answer 1

[Torsten]

https://stackoverflow.com/questions/48200261/fit-plane-to-n-dimensional-points-in-matlab

[Rahul]

Thanks for this. I ended up using SVD instead of PCA. I think, in the use case that I'm describing, either procedure should be fine.