# Decomposing SVD of this dataset by hand (perspective)

I have the following data matrix

$$\left[\begin{array}{ccccc} 1 & 1 & 1 & 0 & 0\\ -3 & -3 & -3 & 0 & 0\\ 2 & 2 & 2 & 0 & 0\\ 0 & 0 & 0 & -1 & -1\\ 0 & 0 & 0 & 2 & 2\\ 0 & 0 & 0 & -1 & -1 \end{array}\right]$$

where the rows are data points and columns dimensions

I am asked to optimize by hand the SVD parameters.

Apparently the answer is this but I have no idea the intuition to get here by hand

$$\left[\begin{array}{cc} a & 0\\ -3a & 0\\ 2a & 0\\ 0 & b\\ 0 & -2b\\ 0 & b \end{array}\right]*\left[\begin{array}{cc} \alpha_{1} & 0\\ 0 & \alpha_{2} \end{array}\right]*\left[\begin{array}{ccccc} c & c & c & 0 & 0\\ 0 & 0 & 0 & d & d \end{array}\right]$$

How do I solve this problem? I can see the first 3 dimensions are collinear, and the last 2 dimensions collinear amongst the points

I am also told to solve for all those values.

• Hint: the original matrix has a clear block form. Each of the nonzero blocks is, just as clearly, the outer product of two vectors. – whuber Dec 3 '18 at 14:54