Diagonally dominant matrix

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In mathematics, a matrix is said to be diagonally dominant if in every row of the matrix, the magnitude of the diagonal entry in that row is larger than the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, if the entry in the ith row and jth column of a matrix is aij, and if

|a_{ii}| > \sum_{j\neq i} |a_{ij}|, \,

then the matrix is diagonally dominant.

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[edit] Variations

The definition in the first paragraph sums entries across rows. It is therefore sometimes called row diagonal dominance. If one changes the definition to sum down columns, this is called column diagonal dominance.

The definition in the first paragraph uses a strict inequality. It is therefore sometimes called strict diagonal dominance. If a weak inequality (\geq) is used, this is called weak diagonal dominance.

If an irreducible matrix is weakly diagonally dominant, but in at least one row (or column) is strictly diagonally dominant, then the matrix is irreducibly diagonally dominant.

[edit] Applications and properties

By the Gershgorin circle theorem, a strictly (or irreducibly) diagonally dominant matrix is non-singular. This result is known as the Levy–Desplanques theorem.[1]

A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semi-definite.

No (partial) pivoting is necessary for a strictly column diagonally dominant matrix when performing Gaussian elimination (LU factorization).

The Jacobi and Gauss-Seidel methods for solving a linear system converge if the matrix is strictly (or irreducibly) diagonally dominant.

Many matrices that arise in finite element methods are diagonally dominant.

A slight variation on the idea of diagonal dominance is used to prove that the pairing on diagrams without loops in the Temperley-Lieb algebra is nondegenerate.[2] For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of q appearing in each row appears only on the diagonal. (The evaluations of such a matrix at large values of q are diagonally dominant in the above sense.)

[edit] Links

[edit] Notes

  1. ^ Horn and Johnson, Thm 6.1.10. This result has been independently rediscovered dozens of times. A few notable ones are Lévy (1881), Desplanques (1886), Minkowski (1900), Hadamard (1903), Schur, Markov (1908), Rohrbach (1931), Gershgorin (1931), Artin (1932), Ostrowski (1937), and Furtwängler (1936). For a history of this "recurring theorem" see: Taussky, Olga (1949). "A recurring theorem on determinants". American Mathematical Monthly 56: 672–676.  Another useful history is in: Schneider, Hans (1977). "Olga Taussky-Todd's influence on matrix theory and matrix theorists". Linear and Multilinear Algebra 5 (3): 197–224. 
  2. ^ K.H. Ko and L. Smolinski (1991). "A combinatorial matrix in 3-manifold theory". Pacific. J. Math. 149: 319-336. 

[edit] References

  • Gene H. Golub & Charles F. Van Loan. Matrix Computations, 1996. ISBN 0-8018-5414-8
  • Roger A. Horn & Charles R. Johnson. Matrix Analysis, Cambridge University Press, 1985. ISBN 0-521-38632-2 (paperback).
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