Stieltjes matrix - Wikipedia

Inmathematics,particularlymatrix theory,aStieltjes matrix,named afterThomas Joannes Stieltjes,is areal symmetric positive definite matrixwithnonpositive off-diagonalentries. A Stieltjes matrix is necessarily anM-matrix.Everyn×nStieltjes matrix is invertible to a nonsingular symmetricnonnegativematrix, though the converse of this statement is not true in general forn> 2.

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

References

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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.

v t e Matrixclasses
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions oneigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz Positive-definite Stieltjes
Satisfying conditions onproductsorinverses	Congruent IdempotentorProjection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation
Used instatistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used ingraph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
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