Relationship between the Inverses of a Matrix and a Submatrix

@article{Ruiz2016RelationshipBT,
  title={Relationship between the Inverses of a Matrix and a Submatrix},
  author={E. Ju{\'a}rez Ruiz and R. Cortes Maldonado and F. P{\'e}rez Rodr{\'i}guez},
  journal={Computaci{\'o}n y Sistemas},
  year={2016},
  volume={20}
}
A simple and straightforward formula for computing the inverse of a submatrix in terms of the inverse of the original matrix is derived. General formulas for the inverse of submatrices of order 𝑛 − 𝑘 as well as block submatrices are derived. The number of additions (or subtractions) and multiplications (or divisions) on the formula is calculated. A variety of numerical results are shown. 
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References

SHOWING 1-10 OF 26 REFERENCES
Explicit inverse of a generalized Vandermonde matrix
Updating the Inverse of a Matrix
TLDR
The history of these fomulas is presented and various applications to statistics, networks, structural analysis, asymptotic analysis, optimization, and partial differential equations are discussed.
MATRIX INVERSION BY THE ANNIHILATION OF RANK
can be readily verified, and provides a method of finding the inverse of a matrix which differs from a matrix of known inverse by a matrix of raiik unity. Equation (1) was originally derived by…
Matrix Inverse as by-Product of Determinant
The determinant of a given square matrix is obtained as the pr oduct of pivot elements evaluated via the iterative matrix order condensation. It follows as the by-product that the inverse of this…
Perturbed matrix inversion with application to LP simplex method
Preconditioning Sparse Nonsymmetric Linear Systems with the Sherman-Morrison Formula
TLDR
It is shown how the matrix A0-1 - A-1, where A0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form $U\Omega V^T$ using the Sherman--Morrison formula.
The solution of linear systems by using the Sherman–Morrison formula
Matrix inversion by rank annihilation
TLDR
This paper outlines a method which allows one to bring the matrix into memory one column at a time and overlap input time with computing time and the number of computations is as small as any known to the writer.
On Deriving the Inverse of a Sum of Matrices
Available expressions are reviewed and new ones derived for the inverse of the sum of two matrices, one of them being nonsingular. Particular attention is given to $({\text{{\bf A}}} + {\text{{\bf…
A survey of modern algebra
This classic, written by two young instructors who became giants in their field, has shaped the understanding of modern algebra for generations of mathematicians and remains a valuable reference and…
...
...