Exercise 11

Suppose that $A=U\Sigma V^T$ is a singular value decomposition of the $m\times n$ matrix $A$. If $\sigma_1,\ldots,\sigma_r$ are the nonzero singular values, the general form of the matrix $\Sigma$ is $$\Sigma = \begin{bmatrix} \sigma_1 & \ldots & 0 & \ldots & 0 \\ 0 & \ldots & 0 & \ldots & 0 \\ 0 & \ldots & \sigma_r & \ldots & 0 \\ 0 & \ldots & 0 & \ldots & 0 \\ 0 & \vdots & 0 & \vdots & 0 \\ 0 & \ldots & 0 & \ldots & 0 \\ \end{bmatrix}.$$

1. If you know that the columns of $A$ are linearly independent, what more can you say about the form of $\Sigma$?

2. If you know that the columns of $A$ span $\real^m$, what more can you say about the form of $\Sigma$?

3. If you know that the columns of $A$ are linearly independent and span $\real^m$, what more can you say about the form of $\Sigma$?