Vector transformations that lead to the identity matrix

[geert200]

Hi all,

I have a question that seems very simple but I just do not see it;)

Let α denote an r×1 vector with arbitrary entries; I'm trying to construct an 1×r vector m such that αm = I, where I is the r×r identity matrix...

The first question is: is this possible?

I tried the following;

let m = α'(α α')^{-1}, but then the problem is that (α α')^{-1} is not defined (rank 1)
how can I fix this;

Thanks in advance Geert

 

[Dick]

You can't. For any two vectors α and m, αm has rank 1. I has rank r.