When is a series of matrices divergent. How to define divergence in this case?

Question

In Quantum Mechanics we deal with series of operators represented as matrices like

$$e^A = 1+ A + \frac{A^2}{2} + \dots$$

and similarly for $\sin(A) $, etc., where $A$ is a matrix. Now my question is

How to define convergence and divergence of a series of matrices?

Like with real numbers if $u_n $ is the $n$-th term of the series then if it does not tend to $0$ we can definitely say that the series is divergent.

For matrices is it just that the $n$-th term should tend to the $\mathbf{0}$ matrix or something else. Because a series of real numbers is divergent if the sum tends to $\pm \infty$ or doesn't tend to anything.

What about a series of matrices. When is a series of matrices divergent. Are there telescopic or alternating series too. What are the tests to verify convergence.

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Edit: I didn't still understand how do we test the convergence for the series of matrices. Are there any standard tests. Can anybody let me know which is one is used to test the convergence of $\exp(A)$, $\sin(A)$, i.e., the one test which is frequently used. Or at least a test to test straight forward divergence

Answer 1

The space of $d\times d$ matrices over $\mathbb{R}$ or $\mathbb{C}$ is typically equipped with the topology such that:

• A sequence of matrices $ A^{(n)} = [a_{ij}^{(n)}]_{1 \leq i,j \leq d}$ converges as $n\to\infty$ if and only if each $ a_{ij}^{(n)}$ converges as $n\to\infty$ for each pair of $i, j$.

So you may apply any of the famous convergence tests to each of the entry. Alternatively, this topology is realized by any of the matrix norm $\| \cdot \|$ such as the Frobenius norm

$$ \| A \|_{F} := \sqrt{\sum_{i,j=1}^{d} |a_{ij}|^2} $$

or the operator norm

$$ \|A \| := \sup_{v : \|v\| = 1} \| Av \|. $$

Then many of the convergence test for the series of real/complex numbers continues to apply if the role of absolute value is replaced by any of the matrix norm:

• If $\sum_{n=1}^{\infty} \| A_n \| < \infty$, then $\sum_{n=1}^{\infty} A_n$ converges.

• If $\limsup_{n\to\infty}\|A_n\|^{1/n} < 1$, then $\sum_{n=1}^{\infty} A_n$ converges.

• If $\limsup_{n\to\infty}\frac{\|A_{n+1}\|}{\|A_n\|} < 1$, then $\sum_{n=1}^{\infty} A_n$ converges.

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However, many of the operators appearing in quantum mechanics are unbounded operators on infinite-dimensional spaces, which necessitates knowledge on the field of functional analysis.

Answer 2

Convergence of a series of matrices means convergence of all entries.

It is standard to consider holomorphic functions of matrices $f(A)$.

But if your interest is quantum mechanics, you usually need to consider unbounded operators and of course we need a different approach in that case.