In mathematics, asquare matrix is said to bediagonally dominant if, for every row of the matrix, the magnitude of the diagonal entry in a row is greater than or equal to the sum of the magnitudes of all the other (off-diagonal) entries in that row. More precisely, the matrix${\displaystyle A}$ is diagonally dominant if

${\displaystyle |a_{ii}|\ge \sum _{j\ne i}|a_{ij}|\mathrm{\forall }i}$

where${\displaystyle a_{ij}}$ denotes the entry in the${\displaystyle i}$th row and${\displaystyle j}$th column.

This definition uses a weak inequality, and is therefore sometimes calledweak diagonal dominance. If a strict inequality (>) is used, this is calledstrict diagonal dominance. The unqualified termdiagonal dominance can mean both strict and weak diagonal dominance, depending on the context.^[1]

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

Any strictly diagonally dominant matrix is trivially aweakly chained diagonally dominant matrix. Weakly chained diagonally dominant matrices are non-singular and include the family ofirreducibly diagonally dominant matrices. These areirreducible matrices that are weakly diagonally dominant, but strictly diagonally dominant in at least one row.

The matrix

isweakly diagonally dominant because

${\displaystyle |a_{11}|\ge |a_{12}|+|a_{13}|}$   since  ${\displaystyle |+3|\ge |-2|+|+1|}$

${\displaystyle |a_{22}|\ge |a_{21}|+|a_{23}|}$   since  ${\displaystyle |+3|\ge |+1|+|+2|}$

${\displaystyle |a_{33}|\ge |a_{31}|+|a_{32}|}$   since  ${\displaystyle |+4|\ge |-1|+|+2|}$.

The matrix

isnot diagonally dominant because

  since  

${\displaystyle |b_{22}|\ge |b_{21}|+|b_{23}|}$   since  ${\displaystyle |+3|\ge |+1|+|+2|}$

  since  .

That is, the first and third rows fail to satisfy the diagonal dominance condition.

The matrix

isstrictly diagonally dominant because

${\displaystyle |c_{11}|>|c_{12}|+|c_{13}|}$   since  ${\displaystyle |-4|>|+2|+|+1|}$

${\displaystyle |c_{22}|>|c_{21}|+|c_{23}|}$   since  ${\displaystyle |+6|>|+1|+|+2|}$

${\displaystyle |c_{33}|>|c_{31}|+|c_{32}|}$   since  ${\displaystyle |+5|>|+1|+|-2|}$.

The following results can be proved trivially fromGershgorin's circle theorem. Gershgorin's circle theorem itself has a very short proof.

A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix^[2]) isnon-singular.

AHermitian diagonally dominant matrix${\displaystyle A}$ with real non-negative diagonal entries ispositive semidefinite. This follows from theeigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix is not necessarily positive semidefinite. For example, consider

However, the real parts of its eigenvalues remain non-negative by Gershgorin's circle theorem.

Similarly, a Hermitian strictly diagonally dominant matrix with real positive diagonal entries ispositive definite.

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

TheJacobi andGauss–Seidel methods for solving alinear system converge if the matrix is strictly (or irreducibly) diagonally dominant.

Many matrices that arise infinite 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 theTemperley–Lieb algebra is non-degenerate.^[3] For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of${\displaystyle q}$ appearing in each row appears only on the diagonal. (The evaluations of such a matrix at large values of${\displaystyle q}$ are diagonally dominant in the above sense.)

• trace of asquare matrix

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

• PlanetMath: Diagonal dominance definition
• PlanetMath: Properties of diagonally dominant matrices
• Mathworld

• Numerical linear algebra
• Matrices (mathematics)

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