Proof of bases of fundamental subspaces

Question

Could anyone help me to prove the following theorems? Here SVD means singular value decomposition

i.e. $U$ is a $m\times m$ matrix with $\mathbf u_i=\frac{Av_i}{||Av_i||}$, V is $n\times n$ matrix with $v_i$ is the eigenvector of $A^TA$, and $\Sigma$ is kind of block matrix with left upper block a diagonal matrix with singular value of A and $0$ otherwise.

Answer 1

Sketch of the proof of the first claim:

• $V^\top$ is invertible so $\text{Col}(U\Sigma V^\top) = \text{Col}(U\Sigma)$
• $\Sigma$ is diagonal and the nonzero diagonal entries are the first $r$ diagonal entries. Thus $U\Sigma$ has columns $\sigma_1 u_1, \ldots, \sigma_r u_r$, and then the remaining $m-r$ columns are zero.