An Explicit Formula for the Inverse of Arrowhead and Doubly Arrow Matrices

Salkuyeh, Davod Khojasteh; Beik, Fatemeh Panjeh Ali

doi:10.1007/s40819-018-0527-5

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Arrowhead Systems and Preliminary Tools1

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2023

2024

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“…. , n, and d 1 − α D −1 β = 0, then the inverse of A can be expressed as a diagonal matrix plus a rank-one update [18]:…”

Section: Arrowhead Systems and Preliminary Toolsmentioning

confidence: 99%

On the balanced truncation error bound and sign parameters from arrowhead realizations

Reiter,

Damm,

Embree

et al. 2024

Adv Comput Math

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Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.

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“…. , n, and d 1 − α D −1 β = 0, then the inverse of A can be expressed as a diagonal matrix plus a rank-one update [18]:…”

Section: Arrowhead Systems and Preliminary Toolsmentioning

confidence: 99%