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Matrix Sequences#

Definition: Matrix Sequence

A matrix sequence is a sequence of matrices.

Convergence#

To get a sense of what it means for a matrix sequence \((\boldsymbol{A}_k)_{k \in \mathcal{D}} \subseteq F^{m \times n}\) of \(m\times n\)-matrices to approach a specific matrix \(\boldsymbol{L} \in F^{m \times n}\), we need a way to quantify how different two matrices, i.e. we need to measure the "distance" between them. If the distance between \(\boldsymbol{A}_k\) and \(\boldsymbol{L}\) approaches \(0\), for \(k \to \infty\), then we can say that \(A_k\) approaches \(\boldsymbol{L}\). To quantify this distance we can use a matrix norms. Luckily, the specific choice for a matrix norms is irrelevant because if the distance approaches \(0\) with respect to one matrix norm, then it does so for all matrix norms.

Theorem: Matrix Convergence via Component Conversion

Let \((\boldsymbol{A}_k)_{k \in \mathcal{D}} \subseteq F^{m \times n}\) be a sequence of \(m\times n\)-matrices over the same topological field \(F\).

The limit of \((\boldsymbol{A}_k)_{k \in \mathcal{D}}\) is \(\boldsymbol{L} \in F^{m \times n}\) if and only if the limits of \(\boldsymbol{A}_k\)'s entries are the entries of \(\boldsymbol{L}\):

\[\lim_{k \to \infty} \boldsymbol{A}_k = \boldsymbol{L} \iff \lim_{k \to \infty} (\boldsymbol{A}_k)_{ij} = \boldsymbol{L}_{ij} \text{ for all } 1 \le i \le m \text{ and all } 1 \le j \le n\]
Example

Consider the matrix sequence defined as follows:

\[\boldsymbol{A}_k = \begin{bmatrix} \frac{4k + 1}{2k} & 1 + \frac{(-1)^k}{k} \\ \mathrm{e}^{-3k} & 5\end{bmatrix}\]

It has the following limit:

\[\boldsymbol{L} = \begin{bmatrix}2 & 1 \\ 0 & 5\end{bmatrix}\]
Proof

TODO

Theorem: Matrix Norm Equivalence

Let \((\boldsymbol{A}_k)_{k \in \mathcal{D}} \subseteq F^{m \times n}\) be a sequence of \(m\times n\)-matrices over the same field \(F\), let \(\boldsymbol{L} \in F^{m \times n}\) and let \(N_1\) and \(N_2\) be norms on \(F^{m \times n}\).

The limit of \(N_1(\boldsymbol{A}_k - \boldsymbol{L})\) for \(k \to \infty\) is zero if and only if the limit of \(N_2(\boldsymbol{A}_k - \boldsymbol{L})\) for \(k \to \infty\) is also zero:

\[\lim_{k \to \infty} N_1(\boldsymbol{A}_k - \boldsymbol{L}) = 0 \iff \lim_{k \to \infty} N_2(\boldsymbol{A}_k - \boldsymbol{L}) = 0\]
Proof

TODO

Theorem: Limits via Norms

Let \((\boldsymbol{A}_k)_{k \in \mathcal{D}} \subseteq F^{m \times n}\) be a sequence of \(m\times n\)-matrices over the same topological field \(F\).

The limit of \((\boldsymbol{A}_k)_{k \in \mathcal{D}}\) is \(\boldsymbol{L} \in F^{m \times n}\) if and only if the limit of any norm of the difference between \(\boldsymbol{A}_k\) and \(\boldsymbol{L}\) is zero:

\[\lim_{k \to \infty} ||\boldsymbol{A}_k - \boldsymbol{L}|| = 0\]
Proof

TODO