# Easy decomposition question

1. Apr 28, 2009

### junglebeast

My book says that, given a 4x4 rank-3 matrix Q (also positive-definite) and its spectral decomposition (eg, eigen decomposition),

$$Q = V D V^{-1}$$

it is easy to factor Q as

$$Q = H J H ^\mathsf{T}$$

where

$$J = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$$

Apparently it's so trivial that they didn't waste the space to say how, but I'm not seeing how to do it...