Transformation Matrix: Understanding Its Purpose and Properties

[danago]

Given a transformation matrix T, which maps objects (x,y) to the image (x',y'). The inverse of T will map the image back to the object.

Just wondering, what happens if matrix T is singular i.e. det(T)=0? Then there is no matrix to map the images back to the object.

My teacher said that he thinks a transformation matrix will never be singular, but he wasnt 100% sure, so I am just wanting to confirm.

Thanks,
Dan.

 

[radou]

My teacher said that he thinks a transformation matrix will never be singular, but he wasnt 100% sure, so I am just wanting to confirm.

Thanks,
Dan.

If, by "transformation matrix", you mean the matrix representation of a linear operator, then of course it can be singular. Think of the mapping A : x --> 0, for every x from the domain.

 

[HallsofIvy]

That depends on what you mean by "transformation". The usual definition in Linear Algebra is simply that L(au+ bv)= aL(u)+ bL(v), which includes transformations that do not have inverses.

On the other hand, if you require that the "transformation" change any n-dimensional object to another n-dimensional object, then it is non-singular.
A singular transformation, such as a projection onto a plane or line, will map n-dimensional objects into lower dimensionals objects and has no inverse.