[R] SVD least squares sub-space projection

[José Augusto M. de Andrade Junior]

Hi all,

A good new year for everybody.
Could somebody help me on a question?

The Singular Value Decomposition of a matrix A gives  A = U * D * t(V)

I A is a M X N matrix, U is the left singular matrix  (M X N),  D is a
diagonal singular values matrix (N X N) and V is the transpose right
singular ortogonal matrix  (N X N).

By taking the first l columns of V, with gives a (l X l) matrix, i
know that i than have a sub-space (R^L)of the original (R^M) space.  I
know that this sub-space basis is  optimal in the least squares sense.

The question is:  given one 3-dim space generated by 6 vectors (A is a
6X3 matrix),  i define a 2-dim orthonormal basis by taking the 2 first
columns of  V, how i can then project a new 3-dim vector in this 2-dim
sub-space just defined?

Thanks in advance.

José Augusto M. de Andrade Jr.
Business Adm. Student
University of Sao Paulo - Brazil