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:
b = 0 and c = 0 → gives only the zero matrix, or
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?