40  Some useful matrix relationships

Author

Affiliation

Aditya Mahajan

McGill University

40.1 Matrix identities

1. If \((I + U)\) is invertible, then \[ U(I + U)^{-1} = I - (I + U)^{-1}. \] This can be verified my multiplying both sides with \((I+U)\).

2. Simplified Sherman-Morrison-Woodbudy formula: If \((I + UV)\) or, equivalently, \((I + VU)\) is invertible, then \[ (I + UV)^{-1} = I - U(I + VU)^{-1}V. \] This can be verified by multiplying both sides with \((I + UV)\). This relationship can also be written as: \[ (I + UV)^{-1} = I - UV^{1/2}(I + V^{1/2}UV^{1/2})^{-1}V^{1/2}. \]

3. A slight generalization of the above is: \[ (I + U T^{-1} V)^{-1} = I - U(T + VU)^{-1}V. \]

4. If \((I + UV)\) or, equivalently, \((I + VU)\) is invertible, then \[ V(I + UV)^{-1} = (I + VU)^{-1}V. \] This can be verified by left multiplying by \((I + VU)\) and right multiplying by \((I + UV)\).