Rules for Symmetric and Positive Semidefinite Matrices: Closure, Self-Adjointness, Polarization, Congruence and Cones

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $n,m\in\mathbb{N}$, let $A,B\in\mathcal{S}(n)$, the set of symmetric matrices, let $t\in\mathbb{R}$, let $C\in\mathcal{M}_{n\times m}(\mathbb{R})$ be a real matrix, and let $x,y\in\mathbb{R}^{n}$, Euclidean space. Matrices have entries, entrywise operations and the zero matrix $0$; juxtaposition is the matrix product or the matrix-vector product, ${}^{\top}$ the transpose, $I_{n}$ the identity matrix, $e_{i}$ the standard basis vectors and $\cdot$ the dot product; $C^{\top}AC$ denotes $(C^{\top}A)C\in\mathcal{M}_{m}(\mathbb{R})$.

¶ $0\in\mathcal{S}(n)$ and $I_{n}\in\mathcal{S}(n)$, and $A+B\in\mathcal{S}(n)$, $A-B\in\mathcal{S}(n)$ and $tA\in\mathcal{S}(n)$.

¶ $(Ax)\cdot y=x\cdot(Ay)$.

¶ For every $P\in\mathcal{M}_{n}(\mathbb{R})$ and all $i,j\in[n]$, $P_{ij}=e_{i}\cdot(Pe_{j})$.

¶ If $z\cdot(Az)=z\cdot(Bz)$ for every $z\in\mathbb{R}^{n}$, then $A=B$.

¶ $C^{\top}AC\in\mathcal{S}(m)$, and if $A$ is positive semidefinite, then so is $C^{\top}AC$.

¶ $C^{\top}C\in\mathcal{S}(m)$, and $C^{\top}C$ is positive semidefinite.

¶ If $A$ and $B$ are positive semidefinite and $t\ge0$, then $A+B$ and $tA$ are positive semidefinite.

¶ If $A$ is positive definite, $B$ is positive semidefinite and $t>0$, then $A+B$ and $tA$ are positive definite.

¶ $I_{n}$ is positive definite.

¶ Every positive definite $D\in\mathcal{S}(n)$ is positive semidefinite.