[R] SVD least squares sub-space projection

From: Josť Augusto M. de Andrade Junior <jamaj69_at_gmail.com>
Date: Sat, 5 Jan 2008 21:35:04 -0300

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.

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

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