The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $m,n\in\mathbb{N}$, let $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ be a real matrix with entries $A_{kl}$. Juxtaposition is the matrix product or the matrix-vector product; $A^{\top}$ is the transpose, $\lVert\cdot\rVert$ the operator norm of matrices, $|\cdot|$ the absolute value and $\sqrt{\cdot}$ the square root; sums are sums of real numbers.

¶ Let $i\in[m]$ and $j\in[n]$, let $e_{i}\in\mathbb{R}^{m}$ and $e_{j}\in\mathbb{R}^{n}$ be the standard basis vectors of Euclidean space, and $\cdot$ the dot product on $\mathbb{R}^{m}$. Then $A_{ij}=e_{i}\cdot(Ae_{j})$.

¶ Let $i\in[m]$ and $j\in[n]$. Then $|A_{ij}|\le\lVert A\rVert$.

¶ $\lVert A\rVert\le\sqrt{\sum_{k=1}^{m}\sum_{l=1}^{n}A_{kl}^{2}}$.

¶ $\lVert A\rVert\le\sum_{k=1}^{m}\sum_{l=1}^{n}|A_{kl}|$.

¶ $\lVert A^{\top}\rVert=\lVert A\rVert$.

¶ Let $t\in\mathbb{R}$, and let $tI_{n}$ be the scalar multiple of the identity matrix $I_{n}$. Then $\lVert tI_{n}\rVert=|t|$.

¶ Let $\mathrm{O}(m)$ and $\mathrm{O}(n)$ be the sets of orthogonal matrices. For all $Q\in\mathrm{O}(m)$ and $R\in\mathrm{O}(n)$: $\lVert QA\rVert=\lVert A\rVert=\lVert AR\rVert$.

¶ Let $d\in\mathbb{R}^{n}$, with components $d_{k}$ as in Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §membership, and let $\operatorname{diag}(d)\in\mathcal{M}_{n}(\mathbb{R})$ be the diagonal matrix with diagonal $d$. The set $\{|d_{k}|:k\in[n]\}$ of real numbers has a greatest element, written $\max_{k\in[n]}|d_{k}|$, and $\lVert\operatorname{diag}(d)\rVert=\max_{k\in[n]}|d_{k}|$.