M-matrix
In mathematics, especially linear algebra, an M-matrix is a matrix whose off-diagonal entries are less than or equal to zero and whose eigenvalues have nonnegative real parts. The set of non-singular M-matrices are a subset of the class of P-matrices, and also of the class of inverse-positive matrices. The name M-matrix was seemingly originally chosen by Alexander Ostrowski in reference to Hermann Minkowski, who proved that if a Z-matrix has all of its row sums positive, then the determinant of that matrix is positive.
Characterizations
An M-matrix is commonly defined as follows:Definition: Let be a real Z-matrix. That is, where for all. Then matrix A is also an M-matrix if it can be expressed in the form, where with, for all, where is at least as large as the maximum of the moduli of the eigenvalues of, and is an identity matrix.
For the non-singularity of, according to the Perron–Frobenius theorem, it must be the case that. Also, for a non-singular M-matrix, the diagonal elements of A must be positive. Here we will further characterize only the class of non-singular M-matrices.
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. For example, Plemmons lists 40 such equivalences. These characterizations has been categorized by Plemmons in terms of their relations to the properties of: positivity of principal minors, inverse-positivity and splittings,
stability, and 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 is an arbitrary matrix, and not necessarily a Z-matrix. Here we mention a few characterizations from each category.
Properties
Below, denotes the element-wise order. That is, for any real matrices A, B of size, we write if for all.Let A be a 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.
- 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.
- There exist inverse-positive matrices and with.
- Every regular splitting of A is convergent.
- There exists a positive diagonal matrix D such that 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 matrix W such that is positive definite.
- is non-singular, and is convergent.
- is non-singular, and for, there exists a positive definite symmetric matrix W such that is positive definite.
- There exists with.
- There exists a positive diagonal matrix D such that has all positive row sums.A has all positive diagonal elements, and there exists a positive diagonal matrix D such that is strictly diagonally dominant.A has all positive diagonal elements, and there exists a positive diagonal matrix D such that is strictly diagonally dominant.