The \(QR\) decomposition of a nonsingular \(X\in\mathbb{C}_{n\times n}\) asserts that \(X=QR\), where \(Q\in\mathbb{C}_{n\times n}\) is unitary and \(R\in\mathbb{C}_{n\times n}\) is upper triangular with positive diagonal entries, and the decomposition is unique. Given a matrix \(A\in\mathbb{C}_{n\times n}\), let \(A(i| j)\) denote the submatrix formed by the first \(i\) rows and the first \(j\) columns of \(A\), \(1\leq i,j\leq n\). For \(A,B,X\in\mathrm{GL}_n(\mathbb{C})\), the authors prove the following theorem. Let \(X=Y^{-1}DY\) be the Jordan decomposition of \(X\), where \(D\) is the Jordan form of \(X\), \(\mathrm{diag}\,D=\mathrm{diag}(\lambda_1, \dots, \lambda_n)\) satisfying \(| \lambda_1| \geq\cdots\geq| \lambda_n| \). Then \(\lim_{m\rightarrow\infty}\,a(AX^mB)^{1/m}=\mathrm{diag}(| \lambda_{\omega(1)}| ,\dots,| \lambda_{\omega(n)}| )\), where the permutation \(\omega\) is uniquely determined by \(YB\); also \(\mathrm{rank}\,\omega(i| j)=\mathrm{rank}(YB)(i| j)\) for \(1\leq i,j\leq n\). Further, let \(R_m=[r_{ij}^{(m)}]_{n\times n}\) in the \(QR\) decomposition of \(AX^mB=Q_mR_m\). Then \[ \overline{\lim}_{m\rightarrow\infty}\,| r_{ij}^{(m)}| ^{1/m}\leq\max_{i\leq k\leq j}\{| \lambda_{\omega(k)}| \}=| \lambda_{\min_{i\leq k\leq j}\omega(k)}| ,\quad 1\leq i\leq j\leq n. \]