Stieltjes matrix

In mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative matrix, though the converse of this statement is not true in general for n > 2.

From the above definition, a Stieltjes matrix is a symmetric invertible Z-matrix whose eigenvalues have positive real parts. As it is a Z-matrix, its off-diagonal entries are less than or equal to zero.

See also

• Hurwitz-stable matrix
• Metzler matrix

References

• David M. Young (2003). Iterative Solution of Large Linear Systems. Dover Publications. p. 42. ISBN 0-486-42548-7.
• Anne Greenbaum (1987). Iterative Methods for Solving Linear Systems. SIAM. p. 162. ISBN 0-89871-396-X.