<s>
In	O
mathematics	O
,	O
a	O
Metzler	B-Algorithm
matrix	I-Algorithm
is	O
a	O
matrix	B-Architecture
in	O
which	O
all	O
the	O
off-diagonal	O
components	O
are	O
nonnegative	O
(	O
equal	O
to	O
or	O
greater	O
than	O
zero	O
)	O
:	O
</s>
<s>
Their	O
properties	O
can	O
be	O
derived	O
by	O
applying	O
the	O
properties	O
of	O
nonnegative	B-Algorithm
matrices	I-Algorithm
to	O
matrices	O
of	O
the	O
form	O
M+aI	O
,	O
where	O
M	O
is	O
a	O
Metzler	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
mathematics	O
,	O
especially	O
linear	B-Language
algebra	I-Language
,	O
a	O
matrix	B-Architecture
is	O
called	O
Metzler	O
,	O
quasipositive	O
(	O
or	O
quasi-positive	O
)	O
or	O
essentially	O
nonnegative	O
if	O
all	O
of	O
its	O
elements	O
are	O
non-negative	O
except	O
for	O
those	O
on	O
the	O
main	O
diagonal	O
,	O
which	O
are	O
unconstrained	O
.	O
</s>
<s>
Metzler	O
matrices	O
are	O
also	O
sometimes	O
referred	O
to	O
as	O
-matrices	O
,	O
as	O
a	O
Z-matrix	B-Algorithm
is	O
equivalent	O
to	O
a	O
negated	O
quasipositive	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
exponential	O
of	O
a	O
Metzler	O
(	O
or	O
quasipositive	O
)	O
matrix	B-Architecture
is	O
a	O
nonnegative	B-Algorithm
matrix	I-Algorithm
because	O
of	O
the	O
corresponding	O
property	O
for	O
the	O
exponential	O
of	O
a	O
nonnegative	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
Metzler	B-Algorithm
matrix	I-Algorithm
has	O
an	O
eigenvector	O
in	O
the	O
nonnegative	O
orthant	O
because	O
of	O
the	O
corresponding	O
property	O
for	O
nonnegative	B-Algorithm
matrices	I-Algorithm
.	O
</s>
