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Mathematically, it's x^2 = x, which implies x^n = x for all positive integers n. Nilpotence (x^2 = 0) is also very helpful some times: it's a process which is

Mathematically, it's x^2 = x, which implies x^n = x for all positive integers n. Nilpotence (x^2 = 0) is also very helpful some times: it's a process which is self-reversing. Like the discrete Fourier transform (if you set up the constants properly).

carreau 6y ago
No, you are confusing with involution. 1/x is an involution. Symetries are often involutions. Squaring a upper triangular matrix with 0 on the diagonal is nilpotent. Derivatiting a polynomial of degree N is nilpotent after N iteration.
contravariant 6y ago
Careful, for nilpotence the power doesn't have to be 2. Also you may be confusing it with x^n = 1 (which I'm not sure how to name, 'root of unity' perhaps). This would be the case for the Fourier transform (with n=4). If x^2 = 0 then applying the Fourier transform twice would null your function, which isn't the case.
pdpi 6y ago
X^2 is a weird way to describe it. A function f is idempotent if f(f(x)) = f(x).
- creata 6y ago
  It's not that weird. People often write iterated composition as f^k, and this is especially true with matrices, where composition and multiplication mean the same thing.
- corty 6y ago
  It is quite common in some fields. Operator application is written without parentheses, and functions are a kind of operator. Therefore: f(x) = f(f(x)) = f f x = f^2 x = f x And leaving out the x, because it is just a placeholder anyways: f f = f^2 = f And of course this means that f^n = f because f^n-1 f = f^n-1 by induction.
  - lamontcg 6y ago
    Depends on if you think of functional notation or matrix operators and linear algebra first. There's also dirac bra-ket notation from QM. I tend to reach for the latter two more than the former. Particularly in this case O^2 = O makes more sense to me than f(f(x)) = f(x). And I just naturally think about A B - B A = 0 rather than f(g(x)) - g(f(x)) = 0 for commutativity.
elcomet 6y ago
self-reversing is not nilpotence. In mathematics, a self-reversing function is called an involution, and it's f^2 (or f(f) ) = Id, the identity function. Nilpotence is very different. It says that if you apply your function a certain number of times, you end up with zero no matter what the input is. For example, projection on x axis + 90 ° rotation of a vector is nilpotent.

carreau 6y ago

No, you are confusing with involution. 1/x is an involution. Symetries are often involutions. Squaring a upper triangular matrix with 0 on the diagonal is nilpotent. Derivatiting a polynomial of degree N is nilpotent after N iteration.

contravariant 6y ago

Careful, for nilpotence the power doesn't have to be 2. Also you may be confusing it with x^n = 1 (which I'm not sure how to name, 'root of unity' perhaps). This would be the case for the Fourier transform (with n=4). If x^2 = 0 then applying the Fourier transform twice would null your function, which isn't the case.

pdpi 6y ago

X^2 is a weird way to describe it. A function f is idempotent if f(f(x)) = f(x).

creata 6y ago
It's not that weird. People often write iterated composition as f^k, and this is especially true with matrices, where composition and multiplication mean the same thing.
corty 6y ago
It is quite common in some fields. Operator application is written without parentheses, and functions are a kind of operator. Therefore: f(x) = f(f(x)) = f f x = f^2 x = f x And leaving out the x, because it is just a placeholder anyways: f f = f^2 = f And of course this means that f^n = f because f^n-1 f = f^n-1 by induction.
- lamontcg 6y ago
  Depends on if you think of functional notation or matrix operators and linear algebra first. There's also dirac bra-ket notation from QM. I tend to reach for the latter two more than the former. Particularly in this case O^2 = O makes more sense to me than f(f(x)) = f(x). And I just naturally think about A B - B A = 0 rather than f(g(x)) - g(f(x)) = 0 for commutativity.

elcomet 6y ago

self-reversing is not nilpotence. In mathematics, a self-reversing function is called an involution, and it's f^2 (or f(f) ) = Id, the identity function. Nilpotence is very different. It says that if you apply your function a certain number of times, you end up with zero no matter what the input is. For example, projection on x axis + 90 ° rotation of a vector is nilpotent.