Possible conditions for the sum of two invertible matrices to be invertible

Created in January 06, 2024

2024   ·   theorem   ·   linalgebra

Theorem: If the product \(AB\) is invertible, then both \(A\) and \(B\) are invertible.

Proposition: Let \(A\) and \(B\) be invertible matrices. If one of the following conditions is satisfied, then \(A+B\) is invertible:

  • \(I + A^{-1}B\) is invertible
  • \[sp(A^{-1}B) \notin -1\]
  • \[\rho(A^{-1}B) < 1\]
  • \(A\) and \(B\) are positive-definite