Power Matrix

Definition¹

For a given matrix ▷ eq01◁ ▷ eq02◁, if there exists a positive number ▷ eq04◁ that satisfies ▷ eq03◁, then ▷ eq02◁ is referred to as nilpotent. In this case, ▷ eq06◁ is the zero matrix ▷ eq01◁.

Explanation

Nil means ‘zero’ or ’none.’ The meaning of potent is ‘powerful,’ and it is the root of the word potential. Thus, the term nilpotent can be understood as ‘having the potential/possibility to become ▷ eq08◁.’ In mathematics, ‘power’ signifies exponentiation, and ‘zero’ refers to the number ▷ eq08◁. Hence, the word nilpotent literally means ’to become zero by exponentiation.’

Theorem

[1]: Upper triangular matrices are nilpotent. (The converse is not true)
[2]: For a square matrix ▷ eq10◁, all of its eigenvalues being ▷ eq08◁ is equivalent to ▷ eq02◁ being a nilpotent matrix.
- Therefore, nilpotent matrices do not have an inverse matrix.
[3]: The trace of a nilpotent matrix ▷ eq02◁ is ▷ eq14◁.

Proof

[1]

Shown by mathematical induction.

[2] ²

$(\implies)$

$A$, being a square matrix, can be represented as $A = Q T Q^{\ast}$ with some unitary matrix $Q$ and upper triangular matrix $T$. Since all eigenvalues of $A$ are ▷ eq08◁, $T$ is an upper triangular matrix with all diagonal elements being $0$, and upper triangular matrices are nilpotent matrices, $T$ is a nilpotent matrix for some $k \in \mathbb{N}$ such that $T^{k} = O$. Therefore, at least for $k$, $A^{k} = Q T^{k} Q^{*} = O$, making $A$ a nilpotent matrix as well.

$(\impliedby)$

Let’s say $A$ is a nilpotent matrix. $$ A^{k} = O $$ The determinant of a product equals the product of determinants, $$ (\det(A))^{k} = \det(A^{k}) = \det(O) = 0 \implies \det(A) = 0 $$

■

[3]

Proof is omitted³.