Stieltjes matrix
Encyclopedia
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes
Thomas Joannes Stieltjes
Thomas Joannes Stieltjes was a Dutch mathematician. He was born in Zwolle and died in Toulouse, France. He was a pioneer in the field of moment problems and contributed to the study of continued fractions....
, is a real symmetric positive definite matrix with nonpositive off-diagonal entries. A Stieltjes matrix is necessarily an M-matrix
M-matrix
In mathematics, especially linear algebra, an M-matrix is a Z-matrix with eigenvalues whose real parts are positive. M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices In mathematics, especially linear algebra, an M-matrix is a Z-matrix with...
. Every n×n Stieltjes matrix is invertible to a nonsingular symmetric nonnegative
Nonnegative matrix
A nonnegative matrix is a matrix in which all the elements are equal to or greater than zeroA positive matrix is a matrix in which all the elements are greater than zero...
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.