Proof of Limits of Matrix Operations, and the Closed Complete Sets of Symmetric and of Positive Semidefinite Matrices

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

By Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line, for a sequence in $\mathbb{R}$, convergence in $(\mathbb{R},d_{\mathbb{R}})$ agrees with that of Convergent Sequences of Real Numbers §converges; we pass between the two readings by this clause.

Clause limit-laws. Assume its hypotheses. The maps $k\mapsto A^{(k)}+B^{(k)}$, $k\mapsto t_{k}A^{(k)}$, $k\mapsto A^{(k)}C^{(k)}$, $k\mapsto A^{(k)}x_{k}$ and $k\mapsto(A^{(k)})^{\top}$ are sequences in $\mathcal{M}_{m\times n}(\mathbb{R})$, $\mathcal{M}_{m\times n}(\mathbb{R})$, $\mathcal{M}_{m\times p}(\mathbb{R})$, $\mathbb{R}^{m}$ and $\mathcal{M}_{n\times m}(\mathbb{R})$ (each term typed by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §operations, Product of Real Matrices §product, Matrix-Vector Product §product and Transpose of a Real Matrix §transpose). The entry sequences of these and of the given sequences of matrices, such as $(A^{(k)}_{ij})$, and the component sequences, such as $((x_{k})_{j})$, are sequences in $\mathbb{R}$ (The Sequences k ↦ f_k(p) of a Sequence in ℝ^I, and the Map of Their Limits §sections), as $\mathcal{M}_{r\times s}(\mathbb{R})=\mathbb{R}^{[r]\times[s]}$ for $r,s\in\mathbb{N}$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §matrices and $\mathbb{R}^{\ell}=\mathbb{R}^{[\ell]}$ for $\ell\in\mathbb{N}$ by Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations §space. Let $i\in[m]$, $j\in[n]$ and $q\in[p]$. By The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §convergence for the sizes $m\times n$ and $n\times p$, and by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence §coordinatewise, the sequences $(A^{(k)}_{ij})$, $(B^{(k)}_{ij})$, $(C^{(k)}_{jq})$ and $((x_{k})_{j})$ converge to $A_{ij}$, $B_{ij}$, $C_{jq}$ and $x_{j}$ in $(\mathbb{R},d_{\mathbb{R}})$, hence in the sense of Convergent Sequences of Real Numbers §converges, as does $(t_{k})$ to $t$. In that sense, by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic (sums and products), the entrywise operations of Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations §operations and Transpose of a Real Matrix §transpose,

$$(A^{(k)}+B^{(k)})_{ij}=A^{(k)}_{ij}+B^{(k)}_{ij}\to A_{ij}+B_{ij}=(A+B)_{ij},\qquad (t_{k}A^{(k)})_{ij}=t_{k}A^{(k)}_{ij}\to tA_{ij}=(tA)_{ij},$$ $$\bigl((A^{(k)})^{\top}\bigr)_{ji}=A^{(k)}_{ij}\to A_{ij}=(A^{\top})_{ji};$$

and by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §arithmetic (products) and Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences §finite-sums with the finite set $[n]=\{1,\dots,n\}$ (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), applied to the maps $(j,k)\mapsto A^{(k)}_{ij}C^{(k)}_{jq}$ and $(j,k)\mapsto A^{(k)}_{ij}(x_{k})_{j}$ on $[n]\times\mathbb{N}$ and the maps $j\mapsto A_{ij}C_{jq}$ and $j\mapsto A_{ij}x_{j}$ on $[n]$ (maps given by expressions: those on $[n]\times\mathbb{N}$ by Maps and Relations Given by Formulas §binary, with $[n]$ and $\mathbb{N}$ for its $c$ and $d$ and $\mathbb{R}$ for its $b$, and those on $[n]$ by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol §map-symbol), together with Product of Real Matrices §product and Matrix-Vector Product §product,

$$(A^{(k)}C^{(k)})_{iq}=\sum_{j=1}^{n}A^{(k)}_{ij}C^{(k)}_{jq}\to\sum_{j=1}^{n}A_{ij}C_{jq}=(AC)_{iq},\qquad (A^{(k)}x_{k})_{i}=\sum_{j=1}^{n}A^{(k)}_{ij}(x_{k})_{j}\to\sum_{j=1}^{n}A_{ij}x_{j}=(Ax)_{i}.$$

These convergences hold in $(\mathbb{R},d_{\mathbb{R}})$ as well, for all $i\in[m]$, $j\in[n]$ and $q\in[p]$. By The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §convergence for the sizes $m\times n$, $m\times p$ and $n\times m$, $(A^{(k)}+B^{(k)})$ converges to $A+B$ and $(t_{k}A^{(k)})$ to $tA$ in $(\mathcal{M}_{m\times n}(\mathbb{R}),d_{m\times n})$, $(A^{(k)}C^{(k)})$ to $AC$ in $(\mathcal{M}_{m\times p}(\mathbb{R}),d_{m\times p})$ and $((A^{(k)})^{\top})$ to $A^{\top}$ in $(\mathcal{M}_{n\times m}(\mathbb{R}),d_{n\times m})$, which are the clauses sum, scalar, product and transpose; and by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence §coordinatewise for $\mathbb{R}^{m}$, $(A^{(k)}x_{k})$ converges to $Ax$ in $(\mathbb{R}^{m},d_{E})$, which is the clause vector.

Closed subsets of $\mathcal{M}_{n}(\mathbb{R})$. By The Open Sets of a Metric Space Form a Set, Given by a Predicative Formula §set, $\mathcal{T}_{d_{n}}$ is the set whose elements are the subsets of $\mathcal{M}_{n}(\mathbb{R})$ that are open in $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$, a metric space by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §metric. So, by Closed Subset of a Topological Space §closed, a subset $F$ of $\mathcal{M}_{n}(\mathbb{R})$ is closed in $(\mathcal{M}_{n}(\mathbb{R}),\mathcal{T}_{d_{n}})$ if and only if $\mathcal{M}_{n}(\mathbb{R})\setminus F$ is open in $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$. Open balls $B_{d_{n}}(A,\rho)$ are those of Open Ball in a Metric Space §ball, with members as in Open Ball in a Metric Space §membership. Write $P=\mathcal{S}_{+}(n)$; by The Invertible Matrices and the Positive Semidefinite Matrices Form Sets §psd, a set lies in $P$ if and only if it lies in $\mathcal{S}(n)$ and is positive semidefinite, and $P\subseteq\mathcal{S}(n)$.

Clause symmetric-closed. Let $U=\mathcal{M}_{n}(\mathbb{R})\setminus\mathcal{S}(n)$ and $A\in U$. The choices are made in this order: $A$, then $i$, $j$ and $\rho$, then $B$. By Symmetric Real Matrices and the Set $\mathcal{S}(n)$ §membership there are $i,j\in[n]$ with $A_{ij}\neq A_{ji}$; then $\rho=|A_{ij}-A_{ji}|/2>0$. Let $B\in B_{d_{n}}(A,\rho)$, so that $B\in\mathcal{M}_{n}(\mathbb{R})$ and $d_{n}(A,B)<\rho$. By The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §entry-bound, $|A_{ij}-B_{ij}|\le d_{n}(A,B)<\rho$ and $|B_{ji}-A_{ji}|\le d_{n}(A,B)<\rho$. By the triangle inequality for the metric $d_{\mathbb{R}}$ (Metric Space §triangle, $d_{\mathbb{R}}$ being a metric by Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces §real-line),

$$|A_{ij}-A_{ji}|\le|A_{ij}-B_{ij}|+|B_{ij}-B_{ji}|+|B_{ji}-A_{ji}|<2\rho+|B_{ij}-B_{ji}|=|A_{ij}-A_{ji}|+|B_{ij}-B_{ji}|,$$

so $|B_{ij}-B_{ji}|>0$, $B_{ij}\neq B_{ji}$, and $B\notin\mathcal{S}(n)$ by Symmetric Real Matrices and the Set $\mathcal{S}(n)$ §membership. Hence $B_{d_{n}}(A,\rho)\subseteq U$, so $U$ is open by Open Subset of a Metric Space §open, and $\mathcal{S}(n)$ is closed.

Clause psd-closed. Let $V=\mathcal{M}_{n}(\mathbb{R})\setminus P$ and $A\in V$. The choices are made in this order: $A$, then $x$, $c$ and $\rho'$, then $B$. If $A\notin\mathcal{S}(n)$, then $A\in U$. The set $U$ is open, as shown in the clause symmetric-closed, so some ball $B_{d_{n}}(A,\rho)$ lies in $U$; and $U\subseteq V$, as $P\subseteq\mathcal{S}(n)$; so this ball about $A$ lies in $V$. Otherwise $A\in\mathcal{S}(n)$ is not positive semidefinite, so by Positive Semidefinite and Positive Definite Real Matrices §psd there is $x\in\mathbb{R}^{n}$ with $x\cdot(Ax)<0$. By The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ §cauchy-schwarz, $0<|x\cdot(Ax)|\le\lVert x\rVert\,\lVert Ax\rVert$, so $\lVert x\rVert\neq0$, and $\lVert x\rVert>0$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square §square. Put $c=-x\cdot(Ax)>0$ and $\rho'=c/\lVert x\rVert^{2}>0$. Let $B\in B_{d_{n}}(A,\rho')$. By The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite §second, $x\cdot(Bx)=x\cdot(Ax)+x\cdot(Bx-Ax)$, and by The Cauchy-Schwarz Inequality on $\mathbb{R}^n$ §cauchy-schwarz, The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §vector-bound and The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §metric (Metric Space §symmetric),

$$|x\cdot(Bx-Ax)|\le\lVert x\rVert\,\lVert Bx-Ax\rVert\le\lVert x\rVert\,d_{n}(B,A)\,\lVert x\rVert=\lVert x\rVert^{2}\,d_{n}(A,B)<\lVert x\rVert^{2}\,\rho'=c .$$

So $x\cdot(Bx)<x\cdot(Ax)+c=0$. Hence $B$ is not a positive semidefinite element of $\mathcal{S}(n)$, by Positive Semidefinite and Positive Definite Real Matrices §psd: either $B\notin\mathcal{S}(n)$, or $B\in\mathcal{S}(n)$ and $B$ fails the condition there at $x$. So $B\in V$, and $B_{d_{n}}(A,\rho')\subseteq V$. In both cases a ball about $A$ lies in $V$, so $V$ is open by Open Subset of a Metric Space §open, and $P$ is closed.

Clause symmetric-space. $\mathcal{S}(n)\subseteq\mathcal{M}_{n}(\mathbb{R})$ by Symmetric Real Matrices and the Set $\mathcal{S}(n)$ §set, and $\mathcal{S}(n)$ is closed in $(\mathcal{M}_{n}(\mathbb{R}),\mathcal{T}_{d_{n}})$ by the clause symmetric-closed. The metric space $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$ (The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §metric) is complete by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §complete. By Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets §restriction, $d_{\mathcal{S}(n)}$ is a metric on $\mathcal{S}(n)$, so $(\mathcal{S}(n),d_{\mathcal{S}(n)})$ is a metric space by Metric Space §metric-space; and by Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets §closed-complete it is complete.

Clause psd-space. $P\subseteq\mathcal{S}(n)\subseteq\mathcal{M}_{n}(\mathbb{R})$, and $P$ is closed in $(\mathcal{M}_{n}(\mathbb{R}),\mathcal{T}_{d_{n}})$ by the clause psd-closed. Write $d_{P}=d_{\mathcal{S}_{+}(n)}$, the restriction of $d_{n}$ to $P\times P$. By Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets §restriction, $d_{P}$ is a metric on $P$, so $(P,d_{P})$ is a metric space by Metric Space §metric-space; and since $(\mathcal{M}_{n}(\mathbb{R}),d_{n})$ is complete by The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence §complete, $(P,d_{P})$ is complete by Subsets of a Metric Space: the Restricted Metric, Sequences, Limits in Closed Subsets and Completeness of Closed Subsets §closed-complete.