Definition

Consider A∈ℝrm×mA \in \mathbb{R}_r^{m \times m}, then AA has a unique pseudoinverse A+∈ℝrm×mA^+ \in \mathbb{R}_r^{m \times m}.

Theorem (Penrose, 1956)

For A∈ℝrm×mA \in \mathbb{R}_r^{m \times m}, G=A+G = A^+ if and only if the following hold:

  1. AGA=AAGA = A
  2. GAG=GGAG = G
  3. (AG)T=AG(AG)^T = AG
  4. (GA)T=GA(GA)^T = GA

Furthermore A+A^+ always exists and is unique.

Theorem (Albert, 1972)

Let A∈ℝrm×mA \in \mathbb{R}_r^{m \times m}, then,

A=limδ→0⁡(ATA+δ2I)−1AT=limδ→0⁡AT(AAT+δ2I)−1 \begin{aligned} A & = \lim_{\delta \to 0} (A^T A + \delta^2 I)^{-1} A^T \\ & = \lim_{\delta \to 0} A^T (AA^T + \delta^2 I)^{-1} \end{aligned}

Notes

  • This can be considered a generalization of the (two-sided) inverse of a matrix where AA−1=I=A−1AAA^{-1} = I = A^{-1}A
    • compare also left inverse and right inverse
  • may use SVD to compute it, e.g. for matrix AA if A=UΣVTA = U \Sigma V^T, then A+=VΣ+UTA^+ = V \Sigma^+ U^T

References

  1. https://www.math.ucla.edu/~laub/33a.2.12s/mppseudoinverse.pdf
  2. G. Strang, Linear algebra and its applications, 4th ed., 2011, p. 122.
  3. https://math.stackexchange.com/questions/458404/how-can-we-compute-pseudoinverse-for-any-matrix
  4. https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/0550c89b69c99e97dcbf52074e293308_MIT18_06SCF11_Ses3.8sum.pdf