ThmDex – An index of mathematical definitions, results, and conjectures.

Characteristic polynomial for a complex identity matrix

Formulation 0

Let $I_N \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	\begin{equation} I_N = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & 1 \end{bmatrix} \end{equation}

Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then \begin{equation} \text{Det}(z I_N - I_N) = (z - 1)^N \end{equation}

Formulation 1

Let $I_N \in \mathbb{C}^{N \times N}$ be a D5699: Complex identity matrix.
Let $z \in \mathbb{C}$ be a D1207: Complex number.

Then \begin{equation} \text{Det}(z I_N - I_N) = (z - 1)^N \end{equation}

Proofs

Proof 0

Let $I_N \in \mathbb{C}^{N \times N}$ be a D6159: Complex square matrix such that

(i)	\begin{equation} I_N = \begin{bmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \vdots & \vdots \\ \vdots & \cdots & \ddots & \vdots \\ 0 & \cdots & \cdots & 1 \end{bmatrix} \end{equation}

Let $z \in \mathbb{C}$ be a D1207: Complex number.