H-matrix (iterative method)

For other uses, see H-matrix.

In mathematics, an H-matrix is a matrix whose comparison matrix is an M-matrix. It is useful in iterative methods.

Definition: Let A = (a_ij) be a n × n complex matrix. Then comparison matrix M(A) of complex matrix A is defined as M(A) = α_ij where α_ij = −|A_ij| for all i ≠ j, 1 ≤ i,j ≤ n and α_ij = |A_ij| for all i = j, 1 ≤ i,j ≤ n. If M(A) is a M-matrix, A is a H-matrix.

Invertible H-matrix guarantees convergence of Gauss–Seidel iterative methods.^[1]

See also

• Hurwitz-stable matrix
• P-matrix
• Perron–Frobenius theorem
• Z-matrix
• L-matrix
• M-matrix
• Comparison matrix