Arrowhead-like matrix

In mathematics, an arrowhead-like matrix is a matrix of the form

[\begin{matrix} X_{m \times m} & B_{o, m \times n} & 0_{m \times k} \\ C_{o, n \times m} & D_{n \times n} & C_{v, n \times k} \\ 0_{k \times m} & B_{v, k \times n} & 0_{k \times k} \end{matrix}]

where X is a diagonal matrix.

Inverses

When n ≥ k, the arrowhead-like matrix has its inverse matrix

{[\begin{matrix} X_{m \times m} & B_{o, m \times n} & 0_{m \times k} \\ C_{o, n \times m} & D_{n \times n} & C_{v, n \times k} \\ 0_{k \times m} & B_{v, k \times n} & 0_{k \times k} \end{matrix}]}^{- 1} = [\begin{matrix} X^{- 1} + X^{- 1} B_{o} (𝒟 - 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟) C_{o} X^{- 1} & - X^{- 1} B_{o} (𝒟 - 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟) & - X^{- 1} B_{o} 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} \\ - (𝒟 - 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟) C_{o} X^{- 1} & 𝒟 - 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟 & 𝒟 C_{v} [B_{v} 𝒟 C_{v}]^{- 1} \\ - [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟 C_{o} X^{- 1} & [B_{v} 𝒟 C_{v}]^{- 1} B_{v} 𝒟 & - [B_{v} 𝒟 C_{v}]^{- 1} \end{matrix}]

𝒟 = (D - C_{o} X^{- 1} B_{o})^{- 1}

Special cases:

k = 0:

{[\begin{matrix} X & B_{o} \\ C_{o} & D \end{matrix}]}^{- 1} = [\begin{matrix} X^{- 1} + X^{- 1} B_{o} 𝒟 C_{o} X^{- 1} & - X^{- 1} B_{o} 𝒟 \\ - 𝒟 C_{o} X^{- 1} & 𝒟 \end{matrix}]

k = n:

{[\begin{matrix} X & B_{o} & 0 \\ C_{o} & D & C_{v} \\ 0 & B_{v} & 0 \end{matrix}]}^{- 1} = [\begin{matrix} X^{- 1} & 0 & - X^{- 1} B_{o} B_{v}^{- 1} \\ 0 & 0 & B_{v}^{- 1} \\ - C_{v}^{- 1} C_{o} X^{- 1} & C_{v}^{- 1} & - C_{v}^{- 1} (D - C_{o} X^{- 1} B_{o}) B_{v}^{- 1} \end{matrix}]

This article "Arrowhead-like matrix" is from Wikipedia. The list of its authors can be seen in its historical. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.