A subspace \(\mathcal X \subset \mathbb{C}^n\) is invariant under a matrix \(A\in\mathbb{C}^{n\times n}\) if \[ A\mathcal X \subset \mathcal X\,. \] If the columns \(X\in\mathbb{C}^{n \times k}\) form an orthonormal basis of \(\mathcal X\), then we obtain the existence of \(A_{11}\in\mathbb{C}^{k \times k}\) such that \(AX=XA_{11}\). Using the block Schur decomposition, the matrix \(X\) is extended to a unitary matrix \([X,X_{\perp}]\) such that \[ A[X,X_{\perp}]=[X,X_{\perp}]\left[\begin{matrix} A_{11} & A_{12}\\ 0 & A_{22}\end{matrix}\right]. \] This implies \(\sigma(A)=\sigma(A_{11}) \cup \sigma(A_{22})\), where \(\sigma(\cdot)\) denotes the spectrum of a matrix.
Throughout this paper, it is assumed that \[ \sigma(A_{11}) \cap \sigma(A_{22})=\emptyset . \] This is a necessary and sufficient condition for the Lipschitz continuity of \(\mathcal X\) with respect to perturbations in \(A\).
The authors consider the block triangular matrix \(A\) as follows \[ A=\left[\begin{matrix} A_{11} & A_{12}\\ 0 & A_{22}\end{matrix}\right],\quad A_{11}\in\mathbb C^{k \times k},\quad A_{22}\in\mathbb C^{(n-k)\times(n-k)}, \] such that \(\sigma(A_{11}) \cap \sigma(A_{22})=\emptyset\), then, under suitable conditions, for all \(\epsilon \geq 0\), \[ \sigma_\epsilon(A) \subset \sigma_{g(\epsilon)}(A_{11}) \cup \sigma_{g(\epsilon)}(A_{22})\,, \] where \(g(\epsilon)=\sqrt{\epsilon(\epsilon+\|A_{12}\|_2)}\). Also, the authors investigate the effect of perturbations on invariant subspaces of a nonsymmetric matrix \(A\).