Invariant Subspaces

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A linear transformation f is a transformation maintaining the following two properties:
f(\alpha + \beta) = f(\alpha) + f(\beta)
f(\lambda x) = \lambda f(x)
Any linear transformation can be represented using a matrix multiplication with a vector.
Any linear transformation f : V \rightarrow V 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 x from such a subspace is transformed using f then
f(x)=\lambda x
where \lambda 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 f.
These subspaces are actually Nullspace (f - \lambda_i I)^{e_i} where m(f) = \prod_{i=1}^{k}(f-\lambda_i I)^{e_i} is the minimum polynomial of f.
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, \prod_{i=1}^{k}(f-\lambda_i I)^{e_i} where e_i > 1) then (f-\lambda_i I)^{e_i} is the zero map on that ith invariant subspace. But for any power less than e_i 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 f^m = 0 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.

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