M-matrix
In
Characterizations
An M-matrix is commonly defined as follows:
Definition: Let A be a n × n real Z-matrix. That is, A = (aij) where aij ≤ 0 for all i ≠ j, 1 ≤ i,j ≤ n. Then matrix A is also an M-matrix if it can be expressed in the form A = sI − B, where B = (bij) with bij ≥ 0, for all 1 ≤ i,j ≤ n, where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an identity matrix.
For the
Many statements that are equivalent to this definition of non-singular M-matrices are known, and any one of these statements can serve as a starting definition of a non-singular M-matrix.[3] For example, Plemmons lists 40 such equivalences.[4] These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other even when matrix A is an arbitrary matrix, and not necessarily a Z-matrix. Here we mention a few characterizations from each category.
Equivalences
Below, ≥ denotes the element-wise order (not the usual
Let A be a n × n real Z-matrix, then the following statements are equivalent to A being a non-singular M-matrix:
Positivity of principal minors
- All the principal minors of A are positive. That is, the determinant of each submatrix of A obtained by deleting a set, possibly empty, of corresponding rows and columns of A is positive.
- A + D is non-singular for each nonnegative diagonal matrix D.
- Every real eigenvalue of A is positive.
- All the leading principal minors of A are positive.
- There exist lower and upper triangular matrices L and U respectively, with positive diagonals, such that A = LU.
Inverse-positivity and splittings
- A is inverse-positive. That is, A−1 exists and A−1 ≥ 0.
- A is monotone. That is, Ax ≥ 0 implies x ≥ 0.
- A has a convergent regular splitting. That is, A has a representation A = M − N, where M−1 ≥ 0, N ≥ 0 with M−1N convergent. That is, ρ(M−1N) < 1.
- There exist inverse-positive matrices M1 and M2 with M1 ≤ A ≤ M2.
- Every regular splitting of A is convergent.
Stability
- There exists a positive diagonal matrix D such that AD + DAT is positive definite.
- A is positive stable. That is, the real part of each eigenvalue of A is positive.
- There exists a symmetric positive definite matrixW such that AW + WAT is positive definite.
- A + I is non-singular, and G = (A + I)−1(A − I) is convergent.
- A + I is non-singular, and for G = (A + I)−1(A − I), there exists a positive definite symmetric matrix W such that W − GTWG is positive definite.
Semipositivity and diagonal dominance
- A is semi-positive. That is, there exists x > 0 with Ax > 0.
- There exists x ≥ 0 with Ax > 0.
- There exists a positive diagonal matrix D such that AD has all positive row sums.
- A has all positive diagonal elements, and there exists a positive diagonal matrix D such that AD is strictly diagonally dominant.
- A has all positive diagonal elements, and there exists a positive diagonal matrix D such that D−1AD is strictly diagonally dominant.
Applications
The primary contributions to M-matrix theory has mainly come from mathematicians and economists. M-matrices are used in mathematics to establish bounds on eigenvalues and on the establishment of convergence criteria for
See also
- A is a non-singular weakly weakly chained diagonally dominant L-matrix.
- If A is an M-matrix, then −A is a Metzler matrix.
- A non-singular symmetric M-matrix is sometimes called a Stieltjes matrix.
- Hurwitz matrix
- P-matrix
- Perron–Frobenius theorem
- Z-matrix
- H-matrix
References
- ^ Fujimoto, Takao & Ranade, Ravindra (2004), "Two Characterizations of Inverse-Positive Matrices: The Hawkins-Simon Condition and the Le Chatelier-Braun Principle" (PDF), Electronic Journal of Linear Algebra, 11: 59–65.
- ISBN 0-89871-321-8.
- .
- .
- ISBN 0-444-10038-5.