Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation

(i) This setting extends Euclidean Space and Real Matrices: Standing Notation (Algebra): its conventions and notation are in force. Background: The Earlier and the New Versions of the Definitions of Symmetric and Positive Semidefinite Matrices, the Operator Norm, Matrix Distance, Eigenvalues and Square Roots Agree.

(ii) Symmetric matrices. $\mathcal{S}(p)$ is the set of symmetric real $p\times p$ matrices, the $A\in\mathcal{M}_{p}(\mathbb{R})$ with $A=A^{\top}$ (Symmetric Real Matrices and the Set $\mathcal{S}(n)$ (i)). For $a\in\mathbb{R}$, $-aI_{p}$ denotes $(-a)I_{p}$. Background: Symmetric Real Matrices and the Set $\mathcal{S}(n)$ (iii) and Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones (i).

(iii) The semidefinite ordering. For $A\in\mathcal{S}(p)$, positive semidefinite and positive definite are as defined there, and $\preceq$ is the positive semidefinite ordering on $\mathcal{S}(p)$. Background: The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, in particular The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure (vi), and Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones.

(iv) Norm and distance. For $A\in\mathcal{M}_{p\times q}(\mathbb{R})$, $\lVert A\rVert$ is the operator norm of $A$; the symbol $\lVert\cdot\rVert$ is shared with the Euclidean norm, and the argument, a matrix or a point, decides which is meant. $d_{p\times q}$, with $d_{p}=d_{p\times p}$, is the distance $d_{p\times q}(A,B)=\lVert A-B\rVert$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, and $d_{\mathcal{S}(p)}$ is its restriction to $\mathcal{S}(p)\times\mathcal{S}(p)$, the restricted metric of Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets (i). Convergence in $\mathcal{M}_{p\times q}(\mathbb{R})$ refers to $d_{p\times q}$, and convergence in $\mathcal{S}(p)$ to $d_{\mathcal{S}(p)}$. Background: The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence, in particular The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (i); Limits of Matrix Operations, and the Closed Complete Sets of Symmetric and of Positive Semidefinite Matrices, in particular Limits of Matrix Operations, and the Closed Complete Sets of Symmetric and of Positive Semidefinite Matrices (ix); The Operator Norm of a Symmetric Matrix: Quadratic Forms, Eigenvalues and the Semidefinite Ordering, in particular The Operator Norm of a Symmetric Matrix: Quadratic Forms, Eigenvalues and the Semidefinite Ordering (i); Rules for the Operator Norm: Bound, Least Constant, Definiteness, Homogeneity, Triangle Inequality and Products, The Operator Norm of Entries, Transposes, Multiples of the Identity, Orthogonal Factors and Diagonal Matrices and Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets.

(v) Spectral notation. Eigenvalues and eigenvectors of $A\in\mathcal{M}_{p}(\mathbb{R})$ are as defined there. For a positive semidefinite $A\in\mathcal{S}(p)$, $A^{1/2}$ is its positive semidefinite square root. Background: The Spectral Theorem for Real Symmetric Matrices, Eigenvalues: the Determinant Criterion, Diagonal Matrices, Orthogonal Similarity and Orthogonality of Eigenvectors of Symmetric Matrices, Positive Semidefinite and Positive Definite Matrices by Their Eigenvalues and Every Positive Semidefinite Matrix Has Exactly One Positive Semidefinite Square Root.