<s>
In	O
mathematics	O
,	O
especially	O
linear	B-Language
algebra	I-Language
,	O
an	O
M-matrix	B-Algorithm
is	O
a	O
Z-matrix	B-Algorithm
with	O
eigenvalues	O
whose	O
real	O
parts	O
are	O
nonnegative	O
.	O
</s>
<s>
The	O
set	O
of	O
non-singular	O
M-matrices	B-Algorithm
are	O
a	O
subset	O
of	O
the	O
class	O
of	O
P-matrices	O
,	O
and	O
also	O
of	O
the	O
class	O
of	O
inverse-positive	B-Algorithm
matrices	I-Algorithm
(	O
i.e.	O
</s>
<s>
matrices	O
with	O
inverses	O
belonging	O
to	O
the	O
class	O
of	O
positive	B-Algorithm
matrices	I-Algorithm
)	O
.	O
</s>
<s>
The	O
name	O
M-matrix	B-Algorithm
was	O
seemingly	O
originally	O
chosen	O
by	O
Alexander	O
Ostrowski	O
in	O
reference	O
to	O
Hermann	O
Minkowski	O
,	O
who	O
proved	O
that	O
if	O
a	O
Z-matrix	B-Algorithm
has	O
all	O
of	O
its	O
row	O
sums	O
positive	O
,	O
then	O
the	O
determinant	O
of	O
that	O
matrix	O
is	O
positive	O
.	O
</s>
<s>
An	O
M-matrix	B-Algorithm
is	O
commonly	O
defined	O
as	O
follows	O
:	O
</s>
<s>
Definition	O
:	O
Let	O
be	O
a	O
real	O
Z-matrix	B-Algorithm
.	O
</s>
<s>
Then	O
matrix	O
A	O
is	O
also	O
an	O
M-matrix	B-Algorithm
if	O
it	O
can	O
be	O
expressed	O
in	O
the	O
form	O
,	O
where	O
with	O
,	O
for	O
all	O
,	O
where	O
is	O
at	O
least	O
as	O
large	O
as	O
the	O
maximum	O
of	O
the	O
moduli	O
of	O
the	O
eigenvalues	O
of	O
,	O
and	O
is	O
an	O
identity	O
matrix	O
.	O
</s>
<s>
Also	O
,	O
for	O
a	O
non-singular	O
M-matrix	B-Algorithm
,	O
the	O
diagonal	O
elements	O
of	O
A	O
must	O
be	O
positive	O
.	O
</s>
<s>
Here	O
we	O
will	O
further	O
characterize	O
only	O
the	O
class	O
of	O
non-singular	O
M-matrices	B-Algorithm
.	O
</s>
<s>
Many	O
statements	O
that	O
are	O
equivalent	O
to	O
this	O
definition	O
of	O
non-singular	O
M-matrices	B-Algorithm
are	O
known	O
,	O
and	O
any	O
one	O
of	O
these	O
statements	O
can	O
serve	O
as	O
a	O
starting	O
definition	O
of	O
a	O
non-singular	O
M-matrix	B-Algorithm
.	O
</s>
<s>
(	O
3	O
)	O
stability	O
,	O
and	O
(	O
4	O
)	O
semipositivity	O
and	O
diagonal	B-Algorithm
dominance	I-Algorithm
.	O
</s>
<s>
It	O
makes	O
sense	O
to	O
categorize	O
the	O
properties	O
in	O
this	O
way	O
because	O
the	O
statements	O
within	O
a	O
particular	O
group	O
are	O
related	O
to	O
each	O
other	O
even	O
when	O
matrix	O
is	O
an	O
arbitrary	O
matrix	O
,	O
and	O
not	O
necessarily	O
a	O
Z-matrix	B-Algorithm
.	O
</s>
<s>
Below	O
,	O
denotes	O
the	O
element-wise	O
order	O
(	O
not	O
the	O
usual	O
positive	B-Algorithm
semidefinite	I-Algorithm
order	O
on	O
matrices	O
)	O
.	O
</s>
<s>
Let	O
A	O
be	O
a	O
real	O
Z-matrix	B-Algorithm
,	O
then	O
the	O
following	O
statements	O
are	O
equivalent	O
to	O
A	O
being	O
a	O
non-singular	O
M-matrix	B-Algorithm
:	O
</s>
<s>
There	O
exist	O
inverse-positive	B-Algorithm
matrices	I-Algorithm
and	O
with	O
.	O
</s>
<s>
There	O
exists	O
a	O
symmetric	B-Algorithm
positive	I-Algorithm
definite	I-Algorithm
matrix	O
W	O
such	O
that	O
is	O
positive	O
definite	O
.	O
</s>
<s>
is	O
non-singular	O
,	O
and	O
for	O
,	O
there	O
exists	O
a	O
positive	O
definite	B-Algorithm
symmetric	I-Algorithm
matrix	I-Algorithm
W	O
such	O
that	O
is	O
positive	O
definite	O
.	O
</s>
<s>
A	O
has	O
all	O
positive	O
diagonal	O
elements	O
,	O
and	O
there	O
exists	O
a	O
positive	O
diagonal	O
matrix	O
D	O
such	O
that	O
is	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
A	O
has	O
all	O
positive	O
diagonal	O
elements	O
,	O
and	O
there	O
exists	O
a	O
positive	O
diagonal	O
matrix	O
D	O
such	O
that	O
is	O
strictly	B-Algorithm
diagonally	I-Algorithm
dominant	I-Algorithm
.	O
</s>
<s>
The	O
primary	O
contributions	O
to	O
M-matrix	B-Algorithm
theory	O
has	O
mainly	O
come	O
from	O
mathematicians	O
and	O
economists	O
.	O
</s>
<s>
M-matrices	B-Algorithm
are	O
used	O
in	O
mathematics	O
to	O
establish	O
bounds	O
on	O
eigenvalues	O
and	O
on	O
the	O
establishment	O
of	O
convergence	O
criteria	O
for	O
iterative	B-Algorithm
methods	I-Algorithm
for	O
the	O
solution	O
of	O
large	O
sparse	B-Algorithm
systems	I-Algorithm
of	O
linear	B-Algorithm
equations	O
.	O
</s>
<s>
M-matrices	B-Algorithm
arise	O
naturally	O
in	O
some	O
discretizations	O
of	O
differential	O
operators	O
,	O
such	O
as	O
the	O
Laplacian	O
,	O
and	O
as	O
such	O
are	O
well-studied	O
in	O
scientific	O
computing	O
.	O
</s>
<s>
M-matrices	B-Algorithm
also	O
occur	O
in	O
the	O
study	O
of	O
solutions	O
to	O
linear	B-Algorithm
complementarity	O
problem	O
.	O
</s>
<s>
Linear	B-Algorithm
complementarity	O
problems	O
arise	O
in	O
linear	B-Algorithm
and	O
quadratic	B-Algorithm
programming	I-Algorithm
,	O
computational	B-Algorithm
mechanics	I-Algorithm
,	O
and	O
in	O
the	O
problem	O
of	O
finding	O
equilibrium	O
point	O
of	O
a	O
bimatrix	O
game	O
.	O
</s>
<s>
Lastly	O
,	O
M-matrices	B-Algorithm
occur	O
in	O
the	O
study	O
of	O
finite	O
Markov	O
chains	O
in	O
the	O
field	O
of	O
probability	O
theory	O
and	O
operations	O
research	O
like	O
queuing	O
theory	O
.	O
</s>
<s>
Meanwhile	O
,	O
economists	O
have	O
studied	O
M-matrices	B-Algorithm
in	O
connection	O
with	O
gross	O
substitutability	O
,	O
stability	O
of	O
a	O
general	O
equilibrium	O
and	O
Leontief	O
's	O
input-output	O
analysis	O
in	O
economic	O
systems	O
.	O
</s>
<s>
In	O
engineering	O
,	O
M-matrices	B-Algorithm
also	O
occur	O
in	O
the	O
problems	O
of	O
Lyapunov	O
stability	O
and	O
feedback	O
control	O
in	O
control	O
theory	O
and	O
is	O
related	O
to	O
Hurwitz	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
In	O
computational	O
biology	O
,	O
M-matrices	B-Algorithm
occur	O
in	O
the	O
study	O
of	O
population	O
dynamics	O
.	O
</s>
