# Pseudoinverse of column submatrix, from pseudoinverse of entire matrix.

Hello,

I am working on a numerical method for the least-squares solution of a linear system. I know that I can approximate the solution to $Ax=b$ with $x=A^+b$, where $A^+$ is the Moore-Penrose pseudoinverse of $A$. In my method, I have to solve such a linear system repeatedly, where the matrices I need to (pseudo)invert are column subsets of a common matrix.

So my question is: Suppose I know the pseudoinverse of $A$ ($A$ is not invertible). Suppose $B$ is another matrix, whose columns are columns of $A$ (a subset, but not permuted). Is there an efficient way to compute the pseudoinverse of $B$ from that of $A$?

Thank you.

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How about the case that $B$ is just one column of $A$? – Anand Aug 8 '11 at 18:44
Even for that simple case, I cannot see a relationship between the pseudoinverses of $A$ and $B$. I can certainly compute them, but I don't see any way to get one from the other. – Jason DeVita Aug 8 '11 at 20:01
you're basically asking for the solution $x=(x_1,x_2,...x_N)$ with the constraint $x_1=x_2=...x_n=0$ ($1<n<N$), given the unconstrained solution; this seems hardly possible. – Carlo Beenakker Aug 11 '11 at 14:36