Diagonally Dominant Matrix - Applications and Properties

Applications and Properties

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 semi-definite. (Proof: Connect with the diagonal matrix that contains the diagonal entries of via the curve of non singular matrices . This shows that . Applying this argument to the minors of, the positive definiteness follows by Sylvester's criterion.) If the symmetry requirement is eliminated, such a matrix is not necessarily positive semi-definite (for example, ); however, the real parts of its eigenvalues are non-negative (see Gershgorin's circle theorem).

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. 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.)