Kunze Linear Algebra · Problem 6.4.1

Elementary Canonical Forms

Mathematics — Methods · invariant-subspaces-r2

Problem

Let TT be the linear operator on R2R^2, the matrix of which in the standard ordered basis is A=[1−122].A = \begin{bmatrix} 1 & -1 \\ 2 & 2 \end{bmatrix}. (a) Prove that the only subspaces of R2R^2 invariant under TT are R2R^2 and the zero subspace. (b) If UU is the linear operator on C2C^2, the matrix of which in the standard ordered basis is AA, show that UU has 11-dimensional invariant subspaces.
Intuitive explanation

The key is understanding what invariant subspaces are: subsets closed under the action of the operator. For part (a), a 11-dimensional invariant subspace would be spanned by an eigenvector of AA. Over the reals, AA has no real eigenvalues (check the characteristic polynomial), so no such subspace exists. Part (b) flips the field to complex numbers, where the same matrix now does have complex eigenvalues, giving you the eigenvectors you need to construct invariant subspaces.

Intuitive explanation

The key is understanding what invariant subspaces are: subsets closed under the action of the operator. For part (a), a 11-dimensional invariant subspace would be spanned by an eigenvector of AA. Over the reals, AA has no real eigenvalues (check the characteristic polynomial), so no such subspace exists. Part (b) flips the field to complex numbers, where the same matrix now does have complex eigenvalues, giving you the eigenvectors you need to construct invariant subspaces.

Related theory

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