Loewner order, Hermitian inner product, and operator norm

PDF p. 6: Section 1.2, Notation; PDF p. 10: Section 2, positive-semidefinite convention

Let $V$ be a finite set. A vector is a function $f:V\to \mathbb{C}$; its Hermitian inner product with $g:V\to \mathbb{C}$ and its induced norm are $⟨f,g⟩={\sum }_{x\in V}\overline{f(x)}g(x)$ and $\mathrm{\parallel }f\mathrm{\parallel }=\sqrt{⟨f,f⟩}$. A $V\times V$ matrix $H$ acts by $(Hf)(x)={\sum }_{y\in V}{H}_{xy}f(y)$. Its conjugate transpose has entries $(H^{\ast }{)}_{xy}=\overline{H_{yx}}$; $H$ is Hermitian, or self-adjoint, if $H=H^{\ast }$. A real matrix is symmetric if $H_{xy}=H_{yx}$ for all $x,y$.

A Hermitian matrix $H$ is positive semidefinite if $⟨f,Hf⟩\ge 0$ for every $f\in {\mathbb{C}}^V$. For Hermitian matrices $A,B$ indexed by the same $V$, write $A⪰B$, equivalently $B⪯A$, if $A-B$ is positive semidefinite. This is the Loewner partial order; in particular it applies to real symmetric matrices of the same dimensions.

For any complex $V\times V$ matrix $H$, its operator norm is $\mathrm{\parallel }H\mathrm{\parallel }={\sup }_{f\mathrm{\ne }0}\mathrm{\parallel }Hf\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }f\mathrm{\parallel }$ when $V$ is nonempty. When $V$ is empty, set $\mathrm{\parallel }H\mathrm{\parallel }=0$. The operator norm is defined for arbitrary matrices here, not only Hermitian ones.

Finite vertex Hilbert space in the article convention

Self-adjoint order, norm bounds and diagonal tests

The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖

Partial order and partially ordered set

The vertex-space definition establishes the inner product and induced norm, matrix action, conjugate transpose and Hermitian terminology; it also shows that a Hermitian matrix has real quadratic values and that a real symmetric matrix is Hermitian (Finite vertex Hilbert space in the article convention). Thus the displayed inequality is meaningful. The positive-semidefinite and order conventions agree with Self-adjoint order, norm bounds and diagonal tests.

The order is reflexive because the zero quadratic form is nonnegative, and transitive because quadratic forms add. For antisymmetry, suppose both $A⪰B$ and $B⪰A$ and put $C=A-B$. Then $⟨f,Cf⟩=0$ for all $f$. Testing the coordinate vector $e_x$ gives $C_{xx}=0$; for distinct $x,y$, testing $e_x+e_y$ and $e_x+i{e}_y$ gives respectively $C_{xy}+C_{yx}=0$ and $i{C}_{xy}-i{C}_{yx}=0$. Hence every entry of $C$ is zero and $A=B$. These tests use the vertex inner product and matrix action of Finite vertex Hilbert space in the article convention. Thus reflexivity, transitivity and antisymmetry establish a partial order in the sense of Partial order and partially ordered set.

For nonempty $V$, normalizing each nonzero $f$ to $f\mathrm{/}\mathrm{\parallel }f\mathrm{\parallel }$ makes the displayed supremum equal to the supremum over unit vectors. The matrix action is a linear endomorphism of the finite-dimensional vertex inner-product space (Finite vertex Hilbert space in the article convention). The operator-norm theorem supplies an attained, finite maximum on its unit sphere, so this supremum equals that maximum (The operator norm is 0 on the zero domain and otherwise equals the largest singular value, attained at a right-singular vector). The stated value on the zero space is the operator-norm definition's convention (The operator norm is zero on the zero domain and otherwise is max_‖ v‖=1‖ Tv‖).

The article uses the symmetric-matrix order and operator norm in Section 1.2 and states the Hermitian quadratic-form convention in Section 2. The empty-set norm convention above makes the finite-dimensional definition explicit; it is not attributed to the article.