Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let the absolute value $|x|$ and the square $x^{2}=x\cdot x$ of $x\in\mathbb{R}$ be as in Absolute Value in an Ordered Field §absolute-value and Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices §arithmetic, and let real matrices and their entries $A_{ij}$ be as in Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §matrices and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §entries.

¶ There is exactly one map $a:\mathbb{R}\to\mathbb{R}$ with $a(x)=|x|$ for every $x\in\mathbb{R}$.

¶ There is exactly one map $q:\mathbb{R}\to\mathbb{R}$ with $q(x)=x^{2}$ for every $x\in\mathbb{R}$.

¶ Let $m,n\in\mathbb{N}$ and $A\in\mathcal{M}_{m\times n}(\mathbb{R})$. The composites $a\circ A$ and $q\circ A$, with $a$ and $q$ the maps of the clauses absolute-value and square, are the unique matrices in $\mathcal{M}_{m\times n}(\mathbb{R})$ with entries $|A_{ij}|$ and $A_{ij}^{2}$, respectively, for all $i\in[m]$ and $j\in[n]$.