B.3 Matrix Identities

For any “conformable” matrices $\mathbf{M}$ and $\mathbf{N}$, it is clearly the case that

We now find an alternative expression for $\mathopen{}\mathclose{{}\left(\mathbf{I}+\mathbf{M}\mathbf{N}}\right)^{-1}$ using the above equation and the fact that $\mathopen{}\mathclose{{}\left(\mathbf{I}+\mathbf{M}\mathbf{N}}\right)$ is its inverse:

Finally, let $\mathbf{M}\stackrel{{\scriptstyle\text{set}}}{{=}}\mathbf{A}^{-1}\mathbf{U}$ and $\mathbf{N}\stackrel{{\scriptstyle\text{set}}}{{=}}\mathbf{C}\mathbf{V}$ in the above equation. Then

This final form is known as the Woodbury matrix-inversion lemma. Notice that we have assumed squareness and invertibility for $\mathbf{A}$ and $\mathbf{C}$, but not $\mathbf{U}$ or $\mathbf{V}$.

The special case in which $\mathbf{U}$ and $\mathbf{V}$ are vectors, $\bm{u}$ and $\bm{v}^{\text{T}}$, and (without further loss of generality) $\mathbf{C}=1$, is known as the Sherman-Morrison formula: