Let $x_k \in \mathbb{R}^n$ be a sequence of vectors such that

$$\lim_{k\to\infty} x_k = x \neq 0.$$

Let $A_k \in \mathbb{R}^{n \times n}$ is a sequence of invertible matrices which does not converge to a single value but $A_k x = y$ for all $k$. Let $y_k := A_k x_k$. Can we say that

$$\lim_{k\to\infty} y_k = y$$

My intuitive thought is that $x_k \to x$ regardless of the matrix product, so $y_k \to A_k x = y$. Is my intuition true? If so how can I prove it rigorously?

## Best Answer

No, see for instance $x_k=\begin{pmatrix}1\\ 1/k\end{pmatrix}$ and $A_k=\begin{pmatrix}1&0\\ 0&k\end{pmatrix}$. However, if the sequence of matrices is bounded (in some operator norm), then $$\lVert A_kx_k-y\rVert=\lVert A_k(x_k-x)\rVert \le \lVert A_k\rVert\lVert x_k-x\rVert\le \lVert x_k-x\rVert\sup_h\lVert A_h\rVert\to 0$$