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In mathematics, a square matrix is said to be **diagonally 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 is diagonally dominant if

where denotes the entry in the th row and th column.

This definition uses a weak inequality, and is therefore sometimes called *weak diagonal dominance*. If a strict inequality (>) is used, this is called *strict diagonal dominance*. The unqualified term *diagonal 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 called *row diagonal dominance*. If one changes the definition to sum down each column, this is called *column diagonal dominance*.

Any strictly diagonally dominant matrix is trivially a weakly chained diagonally dominant matrix. Weakly chained diagonally dominant matrices are non-singular and include the family of *irreducibly diagonally dominant* matrices. These are irreducible matrices that are weakly diagonally dominant, but strictly diagonally dominant in at least one row.

The matrix

is *weakly* diagonally dominant because

- since

- since

- since .

The matrix

is *not* diagonally dominant because

- since

- since

- since .

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

The matrix

is *strictly* diagonally dominant because

- since

- since

- since .

The following results can be proved trivially from Gershgorin'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]}) is non-singular.

A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues 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 is positive 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 non-degenerate.^{[3]} For a matrix with polynomial entries, one sensible definition of diagonal dominance is if the highest power of appearing in each row appears only on the diagonal. (The evaluations of such a matrix at large values of are diagonally dominant in the above sense.)

**^**For instance, Horn and Johnson (1985, p. 349) use it to mean weak diagonal dominance.**^**Horn and Johnson, Thm 6.2.27.**^**K.H. Ko and L. Smolinski (1991). "A combinatorial matrix in 3-manifold theory".*Pacific J. Math.***149**: 319–336.

- Golub, Gene H.; Van Loan, Charles F. (1996).
*Matrix Computations*. 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