Matrix Representation
Over a finite dimensional vector space every linear transformation T : V → V can be represented by a matrix once a basis of V has been chosen.
Suppose now W is a T invariant subspace. Pick a basis C = {v1, ..., vk} of W and complete it to a basis B of V. Then, with respect to this basis, the matrix representation of T takes the form:
where the upper-left block T11 is the restriction of T to W.
In other words, given an invariant subspace W of T, V can be decomposed into the direct sum
Viewing T as an operator matrix
it is clear that T21: W → W' must be zero.
Determining whether a given subspace W is invariant under T is ostensibly a problem of geometric nature. Matrix representation allows one to phrase this problem algebraically. The projection operator P onto W, is defined by P(w + w' ) = w, where w ∈ W and w' ∈ W' . The projection P has matrix representation
A straightforward calculation shows that W = Ran P, the range of P, is invariant under T if and only of PTP = TP. In other words, a subspace W being an element of Lat(T) is equivalent to the corresponding projection satisfying the relation PTP = TP.
If P is a projection (i.e. P2 = P), so is 1 - P, where 1 is the identity operator. It follows from the above that TP = PT if and only if both Ran P and Ran (1 - P) are invariant under T. In that case, T has matrix representation
Colloquially, a projection that commutes with T "diagonalizes" T.
Read more about this topic: Invariant Subspace
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