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I am trying to compute the generalized inverse of an arbitrary (finite or infinite dim'l) complex matrix using a least squares method.

Any idea for the finite and infinite cases?

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Finite dimensions

The details of the finite dimensional case are in How does the SVD solve the least squares problem?. The input matrix is $$ \mathbf{A}\in\mathbb{C}^{m\times n}_{\rho} $$ and the pseudoinverse solution is derived.

Specific cases (underdetermined, overdetermined) are derived in What forms does the Moore-Penrose inverse take under systems with full rank, full column rank, and full row rank? and Pseudo-inverse of a matrix that is neither fat nor tall?

The computational process is detailed in What is step by step logic of pinv (pseudoinverse)?

The SVD in connected to the Fundamental Theorem of Linear Algebra in When pseudo inverse and general inverse of a invertible square matrix will be equal or not equal?

The geometry of the pseudoinverse solution is touched on in Least squares and pseudo-inverse

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