Stoquastic
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the class of ''Z''-matrices are those
matrices Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the ...
whose off-diagonal entries are less than or equal to zero; that is, the matrices of the form: :Z=(z_);\quad z_\leq 0, \quad i\neq j. Note that this definition coincides precisely with that of a negated
Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in st ...
or
quasipositive matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in st ...
, thus the term ''quasinegative'' matrix appears from time to time in the literature, though this is rare and usually only in contexts where references to quasipositive matrices are made. The
Jacobian In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to: *Jacobian matrix and determinant (and in particular, the robot Jacobian) *Jacobian elliptic functions *Jacobian variety * Jacobian ideal *Intermediate Jacobian In mat ...
of a competitive dynamical system is a ''Z''-matrix by definition. Likewise, if the Jacobian of a cooperative dynamical system is ''J'', then (−''J'') is a ''Z''-matrix. Related classes are ''L''-matrices, ''M''-matrices, ''P''-matrices, ''Hurwitz'' matrices and ''Metzler'' matrices. ''L''-matrices have the additional property that all diagonal entries are greater than zero. M-matrices have several equivalent definitions, one of which is as follows: a ''Z''-matrix is an ''M''-matrix if it is nonsingular and its inverse is nonnegative. All matrices that are both ''Z''-matrices and ''P''-matrices are nonsingular ''M''-matrices. In the context of
quantum complexity theory Quantum complexity theory is the subfield of computational complexity theory that deals with complexity classes defined using quantum computers, a computational model based on quantum mechanics. It studies the hardness of computational problems ...
, these are referred to as ''stoquastic operators''.


See also

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Hurwitz-stable matrix In mathematics, a Hurwitz-stable matrix, or more commonly simply Hurwitz matrix, is a square matrix whose eigenvalues all have strictly negative real part. Some authors also use the term stability matrix. Such matrices play an important role in c ...
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M-matrix In mathematics, especially linear algebra, an ''M''-matrix is a matrix whose off-diagonal entries are less than or equal to zero (i.e., it is a ''Z''-matrix) and whose eigenvalues have nonnegative real parts. The set of non-singular ''M''-matrices ...
*
Metzler matrix In mathematics, a Metzler matrix is a matrix in which all the off-diagonal components are nonnegative (equal to or greater than zero): : \forall_\, x_ \geq 0. It is named after the American economist Lloyd Metzler. Metzler matrices appear in st ...
*
P-matrix In mathematics, a -matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P_0-matrices, which are the closure of the class of -matrices, with every principal minor \geq 0. Spectra of -matric ...


References

* * * Matrices (mathematics) {{matrix-stub