ThmDex – An index of mathematical definitions, results, and conjectures.
Formulation 0
Let $A \in \mathbb{C}^{3 \times 3}$ be a D6159: Complex square matrix such that
(i) \begin{equation} A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \end{equation}
(ii) \begin{equation} \Lambda = \begin{bmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & \gamma \end{bmatrix} \end{equation}
Then \begin{equation} \Lambda A = \begin{bmatrix} \lambda a & \lambda b & \lambda c \\ \mu d & \mu e & \mu f \\ \gamma g & \gamma h & \gamma i \end{bmatrix} \end{equation}
Proofs
Proof 0
Let $A \in \mathbb{C}^{3 \times 3}$ be a D6159: Complex square matrix such that
(i) \begin{equation} A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \end{equation}
(ii) \begin{equation} \Lambda = \begin{bmatrix} \lambda & 0 & 0 \\ 0 & \mu & 0 \\ 0 & 0 & \gamma \end{bmatrix} \end{equation}
This result is a particular case of R4992: Complex matrix multiplied from the left by a diagonal matrix. $\square$