<s>
In	O
mathematics	O
and	O
multivariate	B-General_Concept
statistics	I-General_Concept
,	O
the	O
centering	B-Algorithm
matrix	I-Algorithm
is	O
a	O
symmetric	B-Algorithm
and	O
idempotent	O
matrix	O
,	O
which	O
when	O
multiplied	O
with	O
a	O
vector	O
has	O
the	O
same	O
effect	O
as	O
subtracting	O
the	O
mean	O
of	O
the	O
components	O
of	O
the	O
vector	O
from	O
every	O
component	O
of	O
that	O
vector	O
.	O
</s>
<s>
where	O
is	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
of	O
size	O
n	O
and	O
is	O
an	O
n-by-n	O
matrix	O
of	O
all	O
1	O
's	O
.	O
</s>
<s>
is	O
symmetric	B-Algorithm
positive	B-Algorithm
semi-definite	I-Algorithm
.	O
</s>
<s>
has	O
a	O
nullspace	B-Algorithm
of	O
dimension	O
1	O
,	O
along	O
the	O
vector	O
.	O
</s>
<s>
is	O
an	O
orthogonal	B-Algorithm
projection	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
That	O
is	O
,	O
is	O
a	O
projection	B-Algorithm
of	O
onto	O
the	O
(	O
n1	O
)	O
-dimensional	O
subspace	O
that	O
is	O
orthogonal	O
to	O
the	O
nullspace	B-Algorithm
.	O
</s>
<s>
Although	O
multiplication	O
by	O
the	O
centering	B-Algorithm
matrix	I-Algorithm
is	O
not	O
a	O
computationally	O
efficient	O
way	O
of	O
removing	O
the	O
mean	O
from	O
a	O
vector	O
,	O
it	O
is	O
a	O
convenient	O
analytical	O
tool	O
.	O
</s>
<s>
The	O
centering	B-Algorithm
matrix	I-Algorithm
provides	O
in	O
particular	O
a	O
succinct	O
way	O
to	O
express	O
the	O
scatter	O
matrix	O
,	O
of	O
a	O
data	O
sample	O
,	O
where	O
is	O
the	O
sample	O
mean	O
.	O
</s>
