User:Leo1pard/sandbox/Arrowhead-like matrix

Arrowhead-like matrix

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


 * $$\begin{bmatrix}X_{m\times m}&B_{o,m\times n}&0_{m\times k}\\C_{o,n\times m}&D_{n\times n}&C_{v,n\times k}\\0_{k\times m}&B_{v,k\times n}&0_{k\times k}\end{bmatrix}$$

where X is a diagonal matrix.

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


 * $$\begin{bmatrix}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{bmatrix}^{-1}=

\begin{bmatrix}X^{-1}+X^{-1}B_o(\mathcal{D}-\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D})C_oX^{-1} &-X^{-1}B_o(\mathcal{D}-\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D}) &-X^{-1}B_o\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1} \\-(\mathcal{D}-\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D})C_oX^{-1} &\mathcal{D}-\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D} &\mathcal{D}C_v[B_v\mathcal{D}C_v]^{-1} \\-[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D}C_oX^{-1} &[B_v\mathcal{D}C_v]^{-1}B_v\mathcal{D} &-[B_v\mathcal{D}C_v]^{-1}\end{bmatrix}$$


 * $$\mathcal{D}=(D-C_oX^{-1}B_o)^{-1}$$

special cases:

k = 0:


 * $$\begin{bmatrix}X&B_o\\C_o&D\end{bmatrix}^{-1}=

\begin{bmatrix}X^{-1}+X^{-1}B_o\mathcal{D}C_oX^{-1}&-X^{-1}B_o\mathcal{D}\\-\mathcal{D}C_oX^{-1}&\mathcal{D}\end{bmatrix}$$

k = n:


 * $$\begin{bmatrix}X&B_o&0\\C_o&D&C_v\\0&B_v&0\end{bmatrix}^{-1}=\begin{bmatrix}X^{-1}&0&-X^{-1}B_oB_v^{-1}

\\0&0&B_v^{-1} \\-C_v^{-1}C_oX^{-1}&C_v^{-1}&-C_v^{-1}(D-C_oX^{-1}B_o)B_v^{-1}\end{bmatrix}$$