This is exercise 4.2.12 of Fundamentals of matrix Computations from Watkins:
Let $A \in\mathbb R^{n\times m}$, $n \geq m$, $\operatorname{rank}(A) = m$ with complete SVD $ A = U\Sigma V^T$, $U\in \mathbb R^{n\times n}$ and $V\in R^{m\times m}$ are orthogonal matrices and $\Sigma\in \mathbb R^{n\times m} = \operatorname{diag}(\sigma_1,\ldots,\sigma_m)$ with singular values $\sigma_1\geq \sigma_2\geq\cdots\geq \sigma_m\geq 0 $.
I need to find the SVD for $A(A^T A)^{-1} A^T$ :
$$A(A^T A)^{-1} A^T = (U \Sigma V^T)(V(\Sigma^T \Sigma)^{-1} V^T)(V \Sigma^T U^T) = U \Sigma(\Sigma^T \Sigma)^{-1} \Sigma^T U^T. $$
Since $\Sigma(\Sigma^T \Sigma)^{-1} \Sigma^T = \operatorname{diag}(\frac{1}{\sigma_1},\ldots,\frac{1}{\sigma_m}) \operatorname{diag}(\sigma_1,\ldots,\sigma_m) = I_n$ , we have that $ A(A^T A)^{-1} A^T = U I_n U^T = I_n$ . But I think I should obtain $V^T$ in the last expression instead of $U^T$.
What did I do wrong ? Any help will be appreciated