Woodbury matrix identity to solve linear system

Question

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.

Answer 1

Apply the Woodbury matrix identity with $$ A = \mathbf A, \quad U = \mathbf Q, \quad C = \mathbf Q^T, \quad V = \mathbf I_n. $$