Would linear algeba be considered idempotent?

Question

Sorry but I'm not a mathematician, so please bear with me. I understand what idempotence is from a communications point of view, and am wondering if it is correct to include linear algebraic formulas as being idempotent. After all, if y= 2x, then for a given input X, you always get the same result. Is y=2x idempotent?

Cheers, Nap

Answer 1

Any function always gives the same value on the same input: $f(a) = f(b)$ whenever $a = b$. Hence, the property you've observed is really just the fact that expressions like "$2x$" can be thought of as a function of $x$.

Idempotence of $f$ is something different. It means that for all $x$, $f(f(x)) = f(x)$.

Some linear operators are idempotent, e.g. the identity operator $I(x) = x$, or the one that always returns the zero vector. More generally, an operator that "projects" a vector onto a subspace, while keeping the coordinates on that subspace's basis unchanged and zeroing the coordinates on the other dimensions, is an idempotent linear operator.

But most operators are not, as in this very example where $f(x) = 2x$. We then have $f(f(x)) = 4x$, so $f$ is not idempotent unless we're in the trivial zero-dimensional vector space with only the zero vector present.