I have the next linear system: $$\left[{\bf A} + {\bf Q} \cdot {\bf Q}^T\right] \cdot {\bf x} = {\bf z}$$
The dimensions are: ${\bf A} \in \mathbb R^{n \times n}$, ${\bf Q} \in \mathbb R^{n \times m}$, and ${\bf z} \in \mathbb R^{n \times 1}$, where $m < n$.
I want to resolve it only with the Woodbury matrix identity that state: $$(A+UVC)^{-1}=A^{-1}-A^{-1}U(C^{-1}+VA^{-1}U)^{-1}VA^{-1}\text.$$
I don't understand how go from my inicial problem to the statement of Woodbury. Does anyone know how applied it. Thanks you.
Sorry any mistake in my writting.
Apply the Woodbury matrix identity with $$ A = \mathbf A, \quad U = \mathbf Q, \quad C = \mathbf Q^T, \quad V = \mathbf I_n. $$