DescriptionGeneralized inverse subspaces-reflexive.svg	English: Visualize a reflexive generalized inverse X of a matrix A, both viewed as linear maps between the vector spaces U and V. X maps A's image to a complemented subspace of its kernel bijectively, and maps a complemented subspace of A's image all to zero. If both complements are the orthogonal complements, then X is the Moore–Penrose inverse. Here, “K∁ is a complemented subspace of K” means that their direct sum spans the whole space, yet they intersect only at the zero vector.
Date	9 December 2025
Source	Own work
Author	David, but not Hilbert
Other versions	non-reflexive inverse (AXA = A) reflexive inverse (AXA = A, XAX = X) left inverse (XA = Id) right inverse (AX = Id)
SVG development InfoField	The SVG code is valid.  This vector image was created with Typst, with source code available on GitHub

The complementary relationships between subspaces can be derived from the reflexivity. ${\displaystyle X}$, as a reflexive generalized inverse of ${\displaystyle A}$, is defined to satisfy ${\displaystyle AXA=A}$ and ${\displaystyle XAX=X}$. The former implies ${\displaystyle \ker A=\ker AXA\subset \ker XA\subset \ker A}$ and ${\displaystyle \mathrm{im}A=\mathrm{im}AXA\subset \mathrm{im}AX\subset \mathrm{im}A}$. Therefore, ${\displaystyle \ker A=\ker XA}$ and ${\displaystyle \mathrm{im}A=\mathrm{im}AX}$, which are equivalent to ${\displaystyle \ker X\cap \mathrm{im}A=\{0\}}$ and ${\displaystyle \ker A+\mathrm{im}X=U}$ respectively. Similarly, ${\displaystyle XAX=X}$ implies ${\displaystyle \ker A\cap \mathrm{im}X=\{0\}}$ and ${\displaystyle \ker X+\mathrm{im}A=V}$. Combining these four propositions, we reach the conclusion.

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