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The exponential of a Metzler (or quasipositive) matrix is a nonnegative matrix because of the corresponding property for the exponential of a nonnegative matrix. This is natural, once one observes that the generator matrices of continuous-time Markov chains are always Metzler matrices, and that probability distributions are always non-negative.
Metzler matrix: A matrix whose off-diagonal entries are non-negative. Monomial matrix: A square matrix with exactly one non-zero entry in each row and column. Synonym for generalized permutation matrix. Moore matrix: A row consists of a, a q, a q², etc., and each row uses a different variable. Nonnegative matrix: A matrix with all nonnegative ...
In mathematics, the class of Z-matrices are those matrices whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: = ();,. Note that this definition coincides precisely with that of a negated Metzler matrix or quasipositive matrix, thus the term quasinegative matrix appears from time to time in the literature, though this is rare and usually only in ...
The permanent of a matrix A is denoted per A, perm A, or Per A, sometimes with parentheses around the argument. Minc uses Per(A) for the permanent of rectangular matrices, and per(A) when A is a square matrix. [2] Muir and Metzler use the notation | + | +. [3]
Let = be an positive matrix: > for ,.Then the following statements hold. There is a positive real number r, called the Perron root or the Perron–Frobenius eigenvalue (also called the leading eigenvalue, principal eigenvalue or dominant eigenvalue), such that r is an eigenvalue of A and any other eigenvalue λ (possibly complex) in absolute value is strictly smaller than r, |λ| < r.
A strictly diagonally dominant matrix (or an irreducibly diagonally dominant matrix [2]) is non-singular. A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix ...
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In mathematics, a nonnegative matrix, written , is a matrix in which all the elements are equal to or greater than zero, that is, ,. A positive matrix is a matrix in which all the elements are strictly greater than zero. The set of positive matrices is the interior of the set of all non-negative matrices.