📂Matrix Algebra

Convergence of Matrices

Definition¹

Let $\left\{ A_{n} \right\}$ denote a sequence of real (or complex) matrices. The statement that $\left\{ A_{n} \right\}$ converges to the matrix $A$ means that each sequence of entries $\left\{ [A_{n}]_{ij} \right\}$ of $A_{n}$ converges to the corresponding entry $[A]_{ij}$ of $A$ (see convergence).

$$ \lim\limits_{n \to \infty} A_{n} = A, \quad \text{if } \lim\limits_{n \to \infty} [A_{n}]_{ij} = [A]_{ij} \quad \forall i,j $$

$$ \left\{ A_{n} \right\} \to A \text{ as } n \to \infty, \quad \text{if } \left\{ [A_{n}]_{ij} \right\} \to [A]_{ij} \text{ as } n \to \infty \quad \forall i,j $$

Remark

In short, componentwise convergence is the convergence of matrices. Because convergence is defined componentwise, the properties of limits of real sequences (see properties of limits of real sequences) carry over unchanged.

Properties

Below, assume $A$ and $B$ are matrices for which addition and multiplication are well-defined in the given context. For $\lim\limits_{n \to \infty} A_{n} = A$ and $\lim\limits_{n \to \infty} B_{n} = B$ the following hold.

(a) $\lim\limits_{n \to \infty} A_{n} = A \iff \| A_{n} - A \| \to 0$

(b) $\lim\limits_{n \to \infty} (A_{n} + B_{n})$

(c) $\lim\limits_{n \to \infty} (A_{n} B_{n}) = \left( \lim\limits_{n \to \infty} A_{n} \right) \cdot \left( \lim\limits_{n \to \infty} B_{n} \right) = AB$

(d) If $\lim\limits_{n \to \infty} A_{n} = A$, then $\lim\limits_{n \to \infty} (A_{n})^{\ast} = A^{\ast}$

Proof

(b)

$$ \begin{align*} \lim\limits_{n \to \infty} [A_{n}+B_{n}]_{ij} &= \lim\limits_{n \to \infty} \left( [A_{n}]_{ij} + [B_{n}]_{ij} \right) \\ &= \lim\limits_{n \to \infty} [A_{n}]_{ij} + \lim\limits_{n \to \infty} [B_{n}]_{ij} \\ &= [A]_{ij} + [B]_{ij} = [A+B]_{ij} \end{align*} $$

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(c)

$$ \begin{align*} \lim\limits_{n \to \infty} [A_{n}B_{n}]_{ij} &= \lim\limits_{n \to \infty} \sum_{k} [A_{n}]_{ik} [B_{n}]_{kj} \\ &= \sum_{k} [A]_{ik} [B]_{kj} \\ &= [AB]_{ij} \end{align*} $$

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(d)

The conjugate transpose is continuous, and by the definition of continuity the following holds.

$$ \lim\limits_{n \to \infty} (A_{n})^{\ast} = \left( \lim\limits_{n \to \infty} A_{n} \right)^{\ast} = A^{\ast} $$

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