Question

∑[infinity]​ₘ₌₀Am​ is absolutely convergent if each of the scalar series ∑[infinity]ₘ₌₀ent ij​(Am​) is absolutely convergent. a) Show that if ∑​[infinity]ₘ₌₀Am​ is absolutely convergent, then it is convergent. b) Show that the matrix series ∑[infinity]ₘ₌₀=​Am​ is absolutely convergent ⇔∑[infinity]​ₘ₌₀∥Am​∥ is an absolutely convergent series of scalars.

Answer 1

If the matrix series is absolutely convergent, then it is convergent. The matrix series is absolutely convergent if and only if the series of scalar norms is absolutely convergent.

Step-by-step explanation:

To show that if ∑∞ₘ₌₀ Am​ is absolutely convergent, then it is convergent, we can use the fact that absolute convergence implies convergence. If each of the scalar series ∑∞ₘ₌₀ ent ij​(Am​) is absolutely convergent, then by definition, the series converges and the terms of the series approach zero. Since ∑∞ₘ₌₀ Am​ can be expressed as a sum of these scalar series, it follows that if the scalar series converge, the matrix series also converges.

To show that the matrix series ∑∞ₘ₌₀ Am​ is absolutely convergent if and only if ∑∞ₘ₌₀ ∥Am∥ is an absolutely convergent series of scalars, we can use the definition of absolute convergence. If both ∥Am∥ and ∥Bm∥ are convergent series of scalars, then ∑∞ₘ₌₀ (∥Am∥+∥Bm∥) is also a convergent series of scalars. If ∑∞ₘ₌₀ Am​ is a convergent series of matrices, then ∥Am∥ is a series of scalars. Therefore, if ∑∞ₘ₌₀ Am​ is absolutely convergent, then ∥Am∥ is an absolutely convergent series of scalars.