Diagonally dominant matrix
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 larger than or equal to the sum of the magnitudes of all the other (non-diagonal) entries in that row. More precisely, the matrix A is diagonally dominant if
where aij denotes the entry in the ith row and jth column.
Note that 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.
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.
Any strictly diagonally dominant matrix is trivially a weakly chained diagonally dominant matrix. Weakly chained diagonally dominant matrices are nonsingular 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 diagonally dominant because
The matrix
is not diagonally dominant because
That is, the first and third rows fail to satisfy the diagonal dominance condition.
The matrix
is strictly diagonally dominant because
A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix) is non-singular. This result is known as the Levy–Desplanques theorem. This can be proved, for strictly diagonal dominant matrices, using the Gershgorin circle theorem.
A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite.
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