<s>
In	O
mathematics	O
(	O
particularly	O
linear	B-Language
algebra	I-Language
)	O
,	O
a	O
shear	B-Algorithm
matrix	I-Algorithm
or	O
transvection	B-Algorithm
is	O
an	O
elementary	O
matrix	B-Architecture
that	O
represents	O
the	O
addition	O
of	O
a	O
multiple	O
of	O
one	O
row	O
or	O
column	O
to	O
another	O
.	O
</s>
<s>
Such	O
a	O
matrix	B-Architecture
may	O
be	O
derived	O
by	O
taking	O
the	O
identity	B-Algorithm
matrix	I-Algorithm
and	O
replacing	O
one	O
of	O
the	O
zero	O
elements	O
with	O
a	O
non-zero	O
value	O
.	O
</s>
<s>
The	O
name	O
shear	B-Algorithm
reflects	O
the	O
fact	O
that	O
the	O
matrix	B-Architecture
represents	O
a	O
shear	B-Algorithm
transformation	I-Algorithm
.	O
</s>
<s>
Geometrically	O
,	O
such	O
a	O
transformation	O
takes	O
pairs	O
of	O
points	O
in	O
a	O
vector	O
space	O
that	O
are	O
purely	O
axially	O
separated	O
along	O
the	O
axis	O
whose	O
row	O
in	O
the	O
matrix	B-Architecture
contains	O
the	O
shear	B-Algorithm
element	O
,	O
and	O
effectively	O
replaces	O
those	O
pairs	O
by	O
pairs	O
whose	O
separation	O
is	O
no	O
longer	O
purely	O
axial	O
but	O
has	O
two	O
vector	O
components	O
.	O
</s>
<s>
Thus	O
,	O
the	O
shear	B-Algorithm
axis	O
is	O
always	O
an	O
eigenvector	O
of	O
.	O
</s>
<s>
This	O
matrix	B-Architecture
shears	B-Algorithm
parallel	O
to	O
the	O
axis	O
in	O
the	O
direction	O
of	O
the	O
fourth	O
dimension	O
of	O
the	O
underlying	O
vector	O
space	O
.	O
</s>
<s>
A	O
shear	B-Algorithm
parallel	O
to	O
the	O
axis	O
results	O
in	O
and	O
.	O
</s>
<s>
In	O
matrix	B-Architecture
form	O
:	O
</s>
<s>
Similarly	O
,	O
a	O
shear	B-Algorithm
parallel	O
to	O
the	O
axis	O
has	O
and	O
.	O
</s>
<s>
In	O
matrix	B-Architecture
form	O
:	O
</s>
<s>
In	O
3D	O
space	O
this	O
matrix	B-Architecture
shear	B-Algorithm
the	O
YZ	O
plane	O
into	O
the	O
diagonal	O
plane	O
passing	O
through	O
these	O
3	O
points	O
:	O
</s>
<s>
The	O
determinant	O
will	O
always	O
be	O
1	O
,	O
as	O
no	O
matter	O
where	O
the	O
shear	B-Algorithm
element	O
is	O
placed	O
,	O
it	O
will	O
be	O
a	O
member	O
of	O
a	O
skew-diagonal	O
that	O
also	O
contains	O
zero	O
elements	O
(	O
as	O
all	O
skew-diagonals	O
have	O
length	O
at	O
least	O
two	O
)	O
hence	O
its	O
product	O
will	O
remain	O
zero	O
and	O
will	O
not	O
contribute	O
to	O
the	O
determinant	O
.	O
</s>
<s>
Thus	O
every	O
shear	B-Algorithm
matrix	I-Algorithm
has	O
an	O
inverse	O
,	O
and	O
the	O
inverse	O
is	O
simply	O
a	O
shear	B-Algorithm
matrix	I-Algorithm
with	O
the	O
shear	B-Algorithm
element	O
negated	O
,	O
representing	O
a	O
shear	B-Algorithm
transformation	I-Algorithm
in	O
the	O
opposite	O
direction	O
.	O
</s>
<s>
In	O
fact	O
,	O
this	O
is	O
part	O
of	O
an	O
easily	O
derived	O
more	O
general	O
result	O
:	O
if	O
is	O
a	O
shear	B-Algorithm
matrix	I-Algorithm
with	O
shear	B-Algorithm
element	O
,	O
then	O
is	O
a	O
shear	B-Algorithm
matrix	I-Algorithm
whose	O
shear	B-Algorithm
element	O
is	O
simply	O
.	O
</s>
<s>
Hence	O
,	O
raising	O
a	O
shear	B-Algorithm
matrix	I-Algorithm
to	O
a	O
power	O
multiplies	O
its	O
shear	B-Algorithm
factor	O
by	O
.	O
</s>
<s>
If	O
is	O
an	O
shear	B-Algorithm
matrix	I-Algorithm
,	O
then	O
:	O
</s>
<s>
the	O
area	O
,	O
volume	O
,	O
or	O
any	O
higher	O
order	O
interior	O
capacity	O
of	O
a	O
polytope	O
is	O
invariant	O
under	O
the	O
shear	B-Algorithm
transformation	I-Algorithm
of	O
the	O
polytope	O
's	O
vertices	O
.	O
</s>
<s>
Two	O
or	O
more	O
shear	B-Algorithm
transformations	I-Algorithm
can	O
be	O
combined	O
.	O
</s>
<s>
which	O
is	O
a	O
positive	B-Algorithm
definite	I-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Shear	B-Algorithm
matrices	O
are	O
often	O
used	O
in	O
computer	O
graphics	O
.	O
</s>
