Invariant Subspace - Matrix Representation

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, ..., v_k} 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

$T = \begin{bmatrix} T_{11} & T_{12} \\ T_{21} & T_{22} \end{bmatrix} : \begin{matrix}W \\ \oplus \\ W' \end{matrix} \rightarrow \begin{matrix}W \\ \oplus \\ W' \end{matrix},$

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

$P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} : \begin{matrix}W \\ \oplus \\ W' \end{matrix} \rightarrow \begin{matrix}W \\ \oplus \\ W' \end{matrix}.$

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

$T = \begin{bmatrix} T_{11} & 0 \\ 0 & T_{22} \end{bmatrix} : \begin{matrix} \mbox{Ran}P \\ \oplus \\ \mbox{Ran}(1-P) \end{matrix} \rightarrow \begin{matrix} \mbox{Ran}P \\ \oplus \\ \mbox{Ran}(1-P) \end{matrix} \;.$

Colloquially, a projection that commutes with T "diagonalizes" T.

Read more about this topic: Invariant Subspace

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