How do I define the norm of an ordered pair?

Question

Let $t\in\mathbb{R}$ and $A\in\mathbb{R}^{n\times n}$. Note that $\mathbb{R}^{n\times n}$ denotes the set of all real square matrices.

The norm of the ordered pair $(t,A)\in\mathbb{R}\times\mathbb{R}^{n\times n}$ can be defined as: $\left\lVert(t,A)\right\rVert=\left\lVert tA \right\rVert_{F}$. Note that $\left\lVert \cdot \right\rVert_{F}$ is the Frobenius norm.

Is the definition above correct? If so, would you mind explaining why?

Thanks in advance! :)

Answer 1

No. It does not define a norm. One reason is that the zero vector in $\mathbb{R}\times{\mathbb{R}}^{n\times n}$ is the ordered pair $(0,O)$ where $O$ is the zero matrix.

Now, for any $t\neq 0$, we have $\parallel (t,O) \parallel=\parallel O \parallel_F=0$, violating the norm axiom $||(t,A)||=0 \Longrightarrow (t,A)=(0,O)$.