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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 matrix
Metzler matrix

References

David M. Young (2003). Iterative Solution of Large Linear Systems. Dover Publications. p. 42. ISBN 0486425487.
Anne Greenbaum (1987). Iterative Methods for Solving Linear Systems. SIAM. p. 162. ISBN 089871396X.

When k = n or n−1 we saw in the previous section that Vk(Fn) is a principal homogeneous space, and therefore diffeomorphic to the corresponding classical group. These are listed in the table at the right.

Mathematics Encyclopedia

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