Читать книгу Numerical Methods in Computational Finance - Daniel J. Duffy - Страница 94

We conclude this section with a list of matrices whose properties are defined by the signs of their off-diagonal elements.

A Metzler matrix A satisfies where

A Z-matrix is a negated Metzler matrix where .

An M-matrix is a Z-matrix with eigenvalues whose real parts are non-negative. An M-matrix can be expressed in the form . The scalar s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is the identity matrix.

Some applications of M-matrices are from mathematics and economics, for example:

 Establish bounds on eigenvalues.

 Convergence criteria for iterative methods.

 Discretisations of PDEs (for example, in combination with exponential fitting) and monotone finite difference schemes.

 Finite Markov chains.

 Population dynamics.

Finally, L-matrices are defined by: