📂Matrix Algebra

Orthogonal Triangular Matrices are Nilpotent

Theorem¹

$n \times n$ Upper triangular matrix $A$ is a nilpotent matrix.

Explanation

The converse is not true. A simple counterexample is when $A = \begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix}$,

$$ A^{2} = \begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix}\begin{bmatrix} 1 & 1 \\ -1 & -1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} $$

The method of proof is identical, so only the case for upper triangular matrices is discussed.

Proof

We prove this by mathematical induction.

• When $n=1$, it holds.

  Let $A$ be a $1 \times 1$ upper triangular matrix. $$ A = \begin{bmatrix} 0 \end{bmatrix} $$ It is trivially nilpotent.

• Assume it holds for $n=k$, then for $n=k+1$, it holds.

  Let $A$ be a $(k+1) \times (k+1)$ upper triangular matrix. Then, for the $k \times k$ upper triangular matrix $B$, $A$ is represented as the following block matrix. $$ A = \begin{bmatrix} B & \begin{bmatrix} a_{1k+1} \\ a_{2k+1} \\ \vdots \\ a_{kk+1} \end{bmatrix} \\ \begin{bmatrix} 0 & 0 & \cdots & 0 \end{bmatrix} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} B & C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} $$ Then, calculating the power of $A$ yields the following. $$ \begin{align*} A^{2} &= \begin{bmatrix} B & C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} \begin{bmatrix} B & C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} B^{2} & BC \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} \\ A^{3} &= \begin{bmatrix} B^{2} & BC \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} \begin{bmatrix} B & C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} = \begin{bmatrix} B^{3} & B^{2}C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} \\ \vdots & \\ A^{p+1} &= \begin{bmatrix} B^{p+1} & B^{p}C \\ O_{1k} & \begin{bmatrix} 0 \end{bmatrix} \end{bmatrix} \end{align*} $$ Let $B^{p} = O_{kk}$. Then since $A^{p+1} = O_{k+1k+1}$, the $n=k+1$ upper triangular matrix is nilpotent.

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