Right multiplication with matrices

Question

If $X=(x_1,x_2,\dots)$ is an infinite real row vector and $A=(a_{ij}),0<i,j< \infty$ is an infinite real matrix, one may or may not be able to define the matrix product $XA$. For which A can one define right multiplication on the space $Z$ defined as $Z=\{(a) \in \mathbb{R}^{\infty}|\mbox{ }a_n=0 \text{ for all but finitely many n} \}$

Hint: You mention that one cannot always define $XA$ in $\mathbb{R}^\omega$, what specifically goes wrong with a naive approach? — Eric Stucky, Oct 11 '13 at 15:50

Eric Stucky · Answer 1 · 2013-10-11T16:20:30.693

1

Hint: You mention that one cannot always define $XA$ in $\mathbb{R}^\omega$; in particular, the product that we want to use $$(XA)_i =\sum_{j\in\mathbb{N}}x_ia_{ij}$$ has a problem. What is that problem, and to what extent does having finitely many nonzero $x_i$ alleviate it?

edited Oct 11 '13 at 16:20

answered Oct 11 '13 at 16:02

Eric Stucky

12,758
3
38
69

Why does this have 3 unexplained downvotes? This is a solid hint. – Jack M Oct 11 '13 at 18:17
In retrospect, perhaps it's because I messed up my indices: the sum should be $(XA)j=\sum_i x_ia{ij}$. – Eric Stucky Dec 11 '16 at 03:42

Right multiplication with matrices

1 Answers1