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[edit]

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_{o}X^{-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_{o}X^{-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_{o}X^{-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_{o}X^{-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_{o}X^{-1}&-X^{-1}B_{o}{\mathcal {D}}\\-{\mathcal {D}}C_{o}X^{-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_{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{bmatrix}}

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