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A linear transformation f is a transformation maintaining the following two properties:

Any linear transformation can be represented using a matrix multiplication with a vector.

Any linear transformation splits the n-dimensional space V into k f-invariant subspaces – direct sum of which returns the original space, V. These subspaces show eigenvector-like property i.e. when any vector from such a subspace is transformed using then

where is a scaler value. Now, when all these f-invariant subspaces are one-dimensional, then the basis vector of each such one-dimensional vector is called an Eigenvector of the transformation .

These subspaces are actually Nullspace where is the minimum polynomial of .

Notice that if the factors of

*m(f)*are linear and distinct then kernel (or Nullspace) of each factor of*m(f)*is a one dimensional subspace (i.e. it has only one vector as basis). This vector is known as the eigenvector of the transformation*f*. If the transformation,*f*is expressed in terms of these eigenvectors (refer to change of basis theorem), it (the transformation*f*) will be turned into a diagonal matrix. That is why, matrices having distinct linear factors in its minimum polynomial is called a diagonalizable matrix.If the factors are not distinct (i.e. like this, where > 1) then is the zero map on that ith invariant subspace. But for any power less than it will not be a zero map. From this phenomena, a new kind of transformation matrix is formulated – nilpotent matrix. It is defined as a matrix (or linear mapping/transformation)

*f,*where for a positive integer m.For any Nilpotent Matrix there exists an ordered basis with respect to which it (the Nilpotent Matrix) becomes a strictly upper triangular matrix.