Elements of a pseudo inverse matrix

Question

I have a $M\times K$ matrix $H$ which its $m$th row and $k$th column element is $h_{m,k}$. If I show the pseudo inverse matrix with $H^{inv}$ then I would like to write the elements of $H^{inv}$ ($h^{inv}_{i,j}$) as a function of $h_{m,k}$s for $k=1,...,K$ and $m=1,...,M$.

Thanks in advance

Answer 1

The matrix $$ \mathbf{A} \in \mathbb{C}^{m\times n}_{\rho} $$ has singular value decomposition of $$ \mathbf{A} = \mathbf{U} \, \Sigma \, \mathbf{V}^{*}. $$ The column vectors of the domain matrices are $$ u_{k} = \sigma^{-1}_{k} \mathbf{A} v_{k}, \qquad v_{k} = \sigma^{-1}_{k} u^{*}_{k} \mathbf{A}, \qquad k = 1, \rho \tag{1} $$

There is no closed form expression of the matrix elements $\mathbf{A}^{+}_{r,c}$.