In general a matrix operating on a vector may reflect, rotate, shear it, or a combination of these. We can deduce the nature of the transformation by inspection of the matrix.
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 the
direction 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