If the determinant of the matrix is positive, it must be a rotation or a combination of a rotation and a shear.
It the determinant of the matrix is 1 and the
columns or rows have length 1 when considered as a vector then it is
a rotation. The matrix (in two dimensions) may be written
We
can find the rotation angle by equating the transformation matrix to
this and solving for
Note
that in general two one equation needs to be solved for each
of
and
to
get the correct answer.
If the determinant of the matrix is negative, then it must be either a reflection or a combination of a reflection and a shear.
It the determinant of the matrix is -1 and the
columns or rows have length 1 when considered as a vector then it is
a reflection. The matrix (in two dimensions) may be written
We
can find the angle
that
the reflection line makes with the
-
axis by equating the transformation matrix to this and solving
forNote
that in general two one equation needs to be solved for each
of
and
to
get the correct answer.
If the matrix scales a vector, it may scale in
thedirection
or the
direction
or both. If it scales in thedirection
only, then it must leave anyvalue
unchanged, so that if the transformation is represented by
then
so
that
Sinceandare
arbitrary, we must have
and
Similarly,
if the matrix represents a scaling in thedirection
only, then
and
Of
course we may multiply matrices representingandscalings
to obtain a matrix that scales in both directions simultaneously.
The area of any shape transformed is related to
the area of the untransformed shape by the determinant of the
transformation matrix,
If
and
are
the original and transformed shapes then