Having a matrix $\mathbf{A} \in \mathcal{C}^{m\times n}$ I solve following least-squares problem $$Re(\mathbf{A}^H \mathbf{A})x=Re(\mathbf{A}^H\mathbf{b}).$$ If the matrix $\mathbf{A}$ was a real matrix, the solution to the equation above could have been written as $$\mathbf{x} = \sum_{i=1}^{rank(\mathbf{A})} \frac{\mathbf{u}_i^T\mathbf{b}}{s_i}\mathbf{v}_i,$$ where $\mathbf{u}_i$ and $\mathbf{v}_i$ are corresponding left and right singular vectors and $s_i$ is an $i$th singular value.My question is whether the solution to the least-squares problem stated in the first equation can be written in a similar way given the SVD of $\mathbf{A} = \mathbf{U}\mathbf{S}\mathbf{V}^H$?I know that one could split matrix $\mathbf{\tilde A} = [Re(\mathbf{A});~Im(\mathbf{A})]$ and equivalently solve the real problem, but the condition to stay within the complex SVD of the original matrix is of main concern here. Thank you.
Complex least-squares problem
linear algebra;least squares