The concept defines the left and right inverses for rectangular matrices with full rank via $A^TA$ or $AA^T$, establishing that these yield identity mappings only when restricted to specific subspaces (column space vs. row space). In cases where null spaces exist, the theory introduces the pseudoinverse ($A^\dagger$), which serves as a unique linear mapping between the row and column spaces while projecting vectors onto those respective subspaces. This concept extends standard inversion theory within linear algebra by providing a generalized solution framework for under-determined or over-determined systems in regression analysis where perfect two-sided inverses are geometrically impossible.
The concept defines the left and right inverses for rectangular matrices with full rank via $A^TA$ or $AA^T$, establishing that these yield identity mappings only when restricted to specific subspace…