u/Background-Cloud-921

I wrote a short, intuition-first explanation here if anyone’s interested

https://medium.com/think-art/square-roots-of-the-zero-matrix-2-2-9ca80acf30d3

I was exploring the equation A^2 = 0 for 2×2 matrices.

Let

A = [a b

c d]

Then :

A^2 = [a^2 + bc ab + bd

ac + cd bc + d^2]

Setting A^2 = 0 gives :

a^2 + bc = 0

b(a + d) = 0

c(a + d) = 0

d^2 + bc = 0

From this, either:

1. b = 0 and c = 0 → gives only the zero matrix, or

2. a + d = 0 → d = -a

Substituting into the first equation :

a^2 + bc = 0 → bc = -a^2

So it seems all solutions are :

A = [ a b

c -a ], where bc = -a^2

Example:

A = [ 2 -1

4 -2 ]

satisfies A^2 = 0.

So this suggests there are infinitely many square roots of the zero matrix.

Is this a complete characterization of all 2×2 matrices satisfying A^2 = 0, or are there any edge cases I’m missing?

I’ve been thinking about the functional equation f(x+1) = f(x) - 1.

At first impression, it seems like the solution should just be linear, i.e. f(x) = -x + C, since the function decreases by 1 for every unit increase in x.

But I came across an argument suggesting the general solution is actually f(x) = -x + g(x), where g(x+1) = g(x), meaning g is 1-periodic.

For example, f(x) = -x + sin(2πx) also satisfies the condition and is differentiable.

So I’m trying to clarify :

• Is f(x) = -x + C only the polynomial solution ?
• Does differentiability restrict the solution further, or not ?
• What additional conditions (if any) force the function to be strictly linear ?

I also wrote a short article breaking this down more clearly
https://medium.com/think-art/solving-a-subtle-functional-equation-358c71ac0d22

Would appreciate a clear explanation of the full solution space and what assumptions matter.