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

Each result cited is universally quantified over the data in its own statement.

We work in Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices. $\mathbb{R}$ is an ordered field by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum §ordered-field, so the results on ordered fields cited below apply to it; its order $\le$ is a total order by Ordered Fields §ordered-field. For every point $y$ of a Euclidean space, $\lVert y\rVert\ge0$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §square. For every real matrix $D$, $\lVert D\rVert\in\mathbb{R}$ by The Operator Norm of a Real Matrix §values, and $\lVert D\rVert\ge0$ by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §nonnegative. The standard basis vectors exist by The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique §basis, with components $(e_{r})_{k}=\delta_{kr}$ by Identity Matrix and Standard Basis Vectors §basis, where $\delta_{kr}=1$ if $k=r$ and $\delta_{kr}=0$ if $k\neq r$, by Identity Matrix and Standard Basis Vectors §identity.

Clause entry-formula. Let $i\in[m]$ and $j\in[n]$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §column, $(Ae_{j})_{k}=A_{kj}$ for every $k\in[m]$. By Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length §coordinates in $\mathbb{R}^{m}$, with $Ae_{j}$ for its $w$, $e_{i}\cdot(Ae_{j})=(Ae_{j})_{i}=A_{ij}$.

Clause entry-bound. Let $i\in[m]$ and $j\in[n]$. By the clause entry-formula, The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ §cauchy-schwarz in $\mathbb{R}^{m}$, Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length §unit, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound,

$$|A_{ij}|=|e_{i}\cdot(Ae_{j})|\le\lVert e_{i}\rVert\,\lVert Ae_{j}\rVert=\lVert Ae_{j}\rVert\le\lVert A\rVert\,\lVert e_{j}\rVert=\lVert A\rVert.$$

Clause frobenius. Let $\Phi_{A}=\sum_{k=1}^{m}\sum_{l=1}^{n}A_{kl}^{2}$. For $k\in[m]$ and $l\in[n]$, $A_{kl}^{2}=A_{kl}\cdot A_{kl}\ge0$ by Rules of Arithmetic and Order in an Ordered Field §squares; so Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §nonnegative, applied for each $k\in[m]$ to the map $l\mapsto A_{kl}^{2}$ from $[n]$ to $\mathbb{R}$, and then to the map $k\mapsto\sum_{l=1}^{n}A_{kl}^{2}$ from $[m]$ to $\mathbb{R}$, gives $\Phi_{A}\ge0$. These are maps: with $q:\mathbb{R}\to\mathbb{R}$ the map $y\mapsto y^{2}$ of Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix §square, the composite $q\circ A\in\mathcal{M}_{m\times n}(\mathbb{R})$ has entries $(q\circ A)_{kl}=A_{kl}^{2}$ by Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix §entrywise, and is a map $[m]\times[n]\to\mathbb{R}$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §matrices; and Maps Given by Iterated Operations: Partial Iterates, Images of Natural Numbers, Powers and Operations over a Finite Set §finite-operation, with the addition of $\mathbb{R}$ for its $\ast$, $[m]$ and $[n]$ for its $A$ and $B$ ($[n]$ being finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §naturals) and $q\circ A$ for its $f$, gives the maps $l\mapsto A_{kl}^{2}$ from $[n]$ to $\mathbb{R}$, for $k\in[m]$, and the map $k\mapsto\sum_{l=1}^{n}A_{kl}^{2}$ from $[m]$ to $\mathbb{R}$. Let $r=\sqrt{\Phi_{A}}$; by The k-th Root and the Square Root of a Nonnegative Real Number §root and The k-th Root and the Square Root of a Nonnegative Real Number §square-root, $r\ge0$ and $r^{2}=\Phi_{A}$. Let $x\in\mathbb{R}^{n}$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §bound,

$$\lVert Ax\rVert^{2}\le\Phi_{A}\lVert x\rVert^{2}=r^{2}\lVert x\rVert^{2}=(r\,\lVert x\rVert)^{2}.$$

As $\lVert Ax\rVert\ge0$ and $r\,\lVert x\rVert\ge0$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §positive-product), Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §monotone-iff, with $2$ for its $n$, gives $\lVert Ax\rVert\le r\,\lVert x\rVert$. By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least with $r$ for its $c$, $\lVert A\rVert\le r=\sqrt{\Phi_{A}}$.

Clause entry-sum. With $a:\mathbb{R}\to\mathbb{R}$ the map $y\mapsto|y|$ of Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix §absolute-value, let $A'=a\circ A$; it lies in $\mathcal{M}_{m\times n}(\mathbb{R})$ and has entries $A'_{kl}=|A_{kl}|$, by Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix §entrywise. So Maps Given by Iterated Operations: Partial Iterates, Images of Natural Numbers, Powers and Operations over a Finite Set §finite-operation, applied as in the clause frobenius with $A'$ for its $f$, gives the maps $l\mapsto|A_{kl}|$ from $[n]$ to $\mathbb{R}$, for $k\in[m]$, and the map $k\mapsto\sum_{l=1}^{n}|A_{kl}|$ of the row sums from $[m]$ to $\mathbb{R}$. Let $S=\sum_{k=1}^{m}\sum_{l=1}^{n}|A_{kl}|$ and $x\in\mathbb{R}^{n}$. Let $k\in[m]$. By Matrix-Vector Product §product, $(Ax)_{k}=\sum_{l=1}^{n}A_{kl}x_{l}$. By Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §triangle and Rules of Arithmetic and Order in an Ordered Field §absolute-value, $|(Ax)_{k}|\le\sum_{l=1}^{n}|A_{kl}|\,|x_{l}|$. For $l\in[n]$, $|x_{l}|\le\lVert x\rVert$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §coordinate, so $|A_{kl}|\,|x_{l}|\le|A_{kl}|\,\lVert x\rVert$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §nonnegative-scaling with $|A_{kl}|\ge0$ for its $z$. By Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §comparison and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §distributive,

$$|(Ax)_{k}|\le\sum_{l=1}^{n}|A_{kl}|\,\lVert x\rVert=\Bigl(\sum_{l=1}^{n}|A_{kl}|\Bigr)\lVert x\rVert.$$

By Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §coordinate in $\mathbb{R}^{m}$, then Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §comparison and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §distributive once more,

$$\lVert Ax\rVert\le\sum_{k=1}^{m}|(Ax)_{k}|\le\sum_{k=1}^{m}\Bigl(\sum_{l=1}^{n}|A_{kl}|\Bigr)\lVert x\rVert=S\,\lVert x\rVert.$$

By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least with $S$ for its $c$, $\lVert A\rVert\le S$.

Clause transpose. The transpose $A^{\top}\in\mathcal{M}_{n\times m}(\mathbb{R})$ is the value of the map of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique §transpose, by Transpose of a Real Matrix §transpose. We first show $\lVert A\rVert\le\lVert A^{\top}\rVert$, for every $m,n\in\mathbb{N}$ and every $A\in\mathcal{M}_{m\times n}(\mathbb{R})$. Let $x\in\mathbb{R}^{n}$ and $w=Ax\in\mathbb{R}^{m}$. By Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §square and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §adjoint, $\lVert w\rVert^{2}=w\cdot w=(Ax)\cdot w=x\cdot(A^{\top}w)$. By Rules of Arithmetic and Order in an Ordered Field §absolute-value (a number is at most its absolute value), The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ §cauchy-schwarz, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound for $A^{\top}$ with Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §nonnegative-scaling ($\lVert x\rVert\ge0$ for its $z$),

$$\lVert w\rVert\,\lVert w\rVert\le|x\cdot(A^{\top}w)|\le\lVert x\rVert\,\lVert A^{\top}w\rVert\le\lVert x\rVert\,\lVert A^{\top}\rVert\,\lVert w\rVert.$$

If $\lVert w\rVert=0$, then $\lVert w\rVert\le\lVert A^{\top}\rVert\,\lVert x\rVert$, the right side being nonnegative by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §positive-product. If $\lVert w\rVert\neq0$, then $0<\lVert w\rVert$, and Rules of Arithmetic and Order in an Ordered Field §order-product with $\lVert w\rVert$ for its $z$ gives $\lVert w\rVert\le\lVert A^{\top}\rVert\,\lVert x\rVert$. In both cases $\lVert Ax\rVert\le\lVert A^{\top}\rVert\,\lVert x\rVert$, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least gives $\lVert A\rVert\le\lVert A^{\top}\rVert$. Applying this to $A^{\top}\in\mathcal{M}_{n\times m}(\mathbb{R})$, with $(A^{\top})^{\top}=A$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §transpose, gives $\lVert A^{\top}\rVert\le\lVert A\rVert$. Hence $\lVert A^{\top}\rVert=\lVert A\rVert$.

Clause identity. Let $t\in\mathbb{R}$. The identity matrix exists by The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique §identity. Let $x\in\mathbb{R}^{n}$. By Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §matrix-vector, $(tI_{n})x=t(I_{n}x)=tx$, so $\lVert(tI_{n})x\rVert=|t|\,\lVert x\rVert$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §homogeneity. By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least with $|t|$ for its $c$, $\lVert tI_{n}\rVert\le|t|$. With $e_{1}\in\mathbb{R}^{n}$ ($1\in[n]$ by The Segments [n]: Each Is a Set, and 1 Lies in [n] for n ∈ ℕ §one), Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length §unit and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound,

$$|t|=|t|\,\lVert e_{1}\rVert=\lVert(tI_{n})e_{1}\rVert\le\lVert tI_{n}\rVert\,\lVert e_{1}\rVert=\lVert tI_{n}\rVert.$$

Hence $\lVert tI_{n}\rVert=|t|$.

Clause orthogonal. Let $Q\in\mathrm{O}(m)$ and $R\in\mathrm{O}(n)$; the products below exist by The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique §product. First, for every $Q'\in\mathrm{O}(m)$ and $A'\in\mathcal{M}_{m\times n}(\mathbb{R})$: for $x\in\mathbb{R}^{n}$, $(Q'A')x=Q'(A'x)$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §matrix-vector, so by Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion §isometry and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound, $\lVert(Q'A')x\rVert=\lVert A'x\rVert\le\lVert A'\rVert\,\lVert x\rVert$, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least gives $\lVert Q'A'\rVert\le\lVert A'\rVert$. $(\ast)$ By Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion §inverse, $Q^{\top}\in\mathrm{O}(m)$, and $Q^{\top}Q=I_{m}$ by Orthogonal Real Matrices §membership; so by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §associative and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §identity, $Q^{\top}(QA)=(Q^{\top}Q)A=I_{m}A=A$. Then $(\ast)$ with $Q'=Q$, $A'=A$, and with $Q'=Q^{\top}$, $A'=QA$, gives $\lVert QA\rVert\le\lVert A\rVert=\lVert Q^{\top}(QA)\rVert\le\lVert QA\rVert$, so $\lVert QA\rVert=\lVert A\rVert$.

Likewise, for every $R'\in\mathrm{O}(n)$ and $A'\in\mathcal{M}_{m\times n}(\mathbb{R})$: for $x\in\mathbb{R}^{n}$, $(A'R')x=A'(R'x)$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §matrix-vector, so by Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound and Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion §isometry, $\lVert(A'R')x\rVert\le\lVert A'\rVert\,\lVert R'x\rVert=\lVert A'\rVert\,\lVert x\rVert$, and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least gives $\lVert A'R'\rVert\le\lVert A'\rVert$. $(\ast\ast)$ By Rules for Orthogonal Matrices: Inverse, Products, Isometry, Columns, Determinant, Orthonormal Extension and Expansion §inverse, $R^{\top}\in\mathrm{O}(n)$ and $RR^{\top}=I_{n}$; so $(AR)R^{\top}=A(RR^{\top})=AI_{n}=A$ by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §associative and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes §identity. Then $(\ast\ast)$ with $A'=A$, $R'=R$, and with $A'=AR$, $R'=R^{\top}$, gives $\lVert AR\rVert\le\lVert A\rVert=\lVert(AR)R^{\top}\rVert\le\lVert AR\rVert$, so $\lVert AR\rVert=\lVert A\rVert$.

Clause diagonal. Let $D=\operatorname{diag}(d)$, let $a:\mathbb{R}\to\mathbb{R}$ be the map $y\mapsto|y|$ of Absolute Value and Squaring as Maps on the Reals, and Entrywise Absolute Values and Squares of a Real Matrix §absolute-value, and let $T=\{|d_{k}|:k\in[n]\}$. Take no variables $x_{i}$ and the guard $d\in\mathbb{R}^{n}$. In it and in the formulas below, $d$, $\mathbb{R}^{n}$, $[n]$, $\mathbb{R}$ and the map $a$ enter as parameters: set variables to which these sets are assigned. Under the guard $d\in\mathbb{R}^{n}\wedge k\in[n]$, for the variable $k$, the term $|d_{k}|=a(d(k))$ is of type $\mathbb{R}$, by Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them §variable for $k$ and Typed Terms under a Predicative Guard: Variables, Parameters, Values of Maps and Ordered Pairs, and Predicative Equations and Memberships between Them §value applied first to the map $d:[n]\to\mathbb{R}$ (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §membership) and then to $a$. The interval $[n]$ is finite by Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §naturals. So, by Images of Typed Terms: the Set of Values $\{t:u\in D\}$ Is a Set, Finite when $D$ Is Finite, and a Typed Term §set, Images of Typed Terms: the Set of Values $\{t:u\in D\}$ Is a Set, Finite when $D$ Is Finite, and a Typed Term §members and Images of Typed Terms: the Set of Values $\{t:u\in D\}$ Is a Set, Finite when $D$ Is Finite, and a Typed Term §finite, with $k$ for its $u$, $[n]$ for its $D$ and $\mathbb{R}$ for its $B$, $T$ is a finite set and a subset of $\mathbb{R}$, and for every set $b$: $b\in T$ if and only if $b=|d_{k}|$ for some $k\in[n]$. $T$ is nonempty since $|d_{1}|\in T$, as $1\in[n]$ by The Segments [n]: Each Is a Set, and 1 Lies in [n] for n ∈ ℕ §one. As $\le$ is a total order on $\mathbb{R}$, Finite Sets: the Pigeonhole Principle, Uniqueness of the Length, Subsets, Unions, Products, Images, Bounded Sets of Natural Numbers, Extreme Elements, Sets of Maps, Finite Unions and Finite Choice §extremes gives a greatest element of $T$, unique by Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least; write $\mu=\max T=\max_{k\in[n]}|d_{k}|$. By Bounds, Least and Greatest Elements, Suprema and Infima for a Partial Order §least, $\mu\in T$, so $\mu=|d_{k_{0}}|$ for some $k_{0}\in[n]$; and $|d_{k}|\le\mu$ for every $k\in[n]$.

Upper bound. Let $x\in\mathbb{R}^{n}$. By Eigenvalues: the Determinant Criterion, Diagonal Matrices, Orthogonal Similarity and Orthogonality of Eigenvectors of Symmetric Matrices §diagonal-algebra, $(Dx)_{k}=d_{k}x_{k}$ for $k\in[n]$. For $k\in[n]$, $0\le|d_{k}|\le\mu$ and $0\le\mu=|d_{k_{0}}|$, so Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §monotone-iff, with $2$ for its $n$, gives $|d_{k}|^{2}\le\mu^{2}$; as $d_{k}^{2}=|d_{k}|^{2}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §square-abs, $d_{k}^{2}\le\mu^{2}$, and $d_{k}^{2}x_{k}^{2}\le\mu^{2}x_{k}^{2}$ by Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §nonnegative-scaling with $x_{k}^{2}\ge0$ (Rules of Arithmetic and Order in an Ordered Field §squares) for its $z$. By Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §square, Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §comparison and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality §distributive,

$$\lVert Dx\rVert^{2}=\sum_{k=1}^{n}d_{k}^{2}x_{k}^{2}\le\sum_{k=1}^{n}\mu^{2}x_{k}^{2}=\mu^{2}\lVert x\rVert^{2}=(\mu\,\lVert x\rVert)^{2}.$$

Since $\lVert Dx\rVert\ge0$ and $\mu\,\lVert x\rVert\ge0$ (Inequalities in an Ordered Field: Mixed Transitivity, Strict Sums, Signs, Products, Natural Numbers, Halving, Reciprocals, Absolute Values and Squares §positive-product), Powers in a Commutative Ring, a Field and an Ordered Field: Exponent Laws, Factorisation, Geometric Sums, Monotonicity and Bernoulli's Inequality §monotone-iff, with $2$ for its $n$, gives $\lVert Dx\rVert\le\mu\,\lVert x\rVert$. By Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §least with $\mu$ for its $c$, $\lVert D\rVert\le\mu$.

Lower bound. Let $e=e_{k_{0}}\in\mathbb{R}^{n}$. For $k\in[n]$, $(De)_{k}=d_{k}\delta_{kk_{0}}$ by Eigenvalues: the Determinant Criterion, Diagonal Matrices, Orthogonal Similarity and Orthogonality of Eigenvectors of Symmetric Matrices §diagonal-algebra, and $(d_{k_{0}}e)_{k}=d_{k_{0}}\delta_{kk_{0}}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §operations; both are $d_{k_{0}}$ if $k=k_{0}$ and $0$ if $k\neq k_{0}$. So $De=d_{k_{0}}e$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §equality. By Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §homogeneity, Standard Basis Vectors of ℝ^q: Coordinates, Orthonormality and Unit Length §unit and Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products §bound,

$$\mu=|d_{k_{0}}|\,\lVert e\rVert=\lVert De\rVert\le\lVert D\rVert\,\lVert e\rVert=\lVert D\rVert.$$

Hence $\lVert\operatorname{diag}(d)\rVert=\mu=\max_{k\in[n]}|d_{k}|$.