Idempotent and nilpotent matrices are defined differently. Why?

Question

We call $A$ idempotent if $A^2$ is $A$. But we call A nilpotent if $A^k$ is $0$ for some integer $k$. Why are not they defined uniformly like both with power 2 or both with power some integer $k$.

Answer 1

The root potent in each term refers to (integer) powers. Idem means "same", while nil refers to "zero". In this sense, the terms are self-descriptive:

• Idempotent means "the second power of $A$ (and hence every higher integer power) is equal to $A$".

• Nilpotent means "some power of $A$ is equal to the zero matrix".

As Bernard suggests, definitions are made for convenience of use. Both of the preceding occur often enough to deserve a special term.

In practice, the condition $A^{2} = A$ is far more important than:

• For some integer $k > 2$, $A^{k} = A$ and $A^{j} \neq A$ for $2 \leq j < k$.

Though this "$k$-idempotent" condition may well be useful in some parts of mathematics, I can't recall ever seeing it "in the wild". By contrast, idempotent operators model projections (as Bernard notes), or operations (like pressing the call button of an elevator) that "have no additional effect when repeated".

Incidentally, the eigenvalues of an idempotent matrix are all $0$ or $1$ (and the domain decomposes as a direct sum of eigenspaces), while the eigenvalues of a nilpotent matrix are all $0$ (and a non-zero nilpotent matrix is never diagonalizable). By contrast, the eigenvalues of a real "$k$-idempotent" matrix ($k > 3$) are not generally real.