Proof of Limits and Bounded Sequences of Symmetric Real Matrices

Throughout we use the notation of the statement. We first record three elementary facts.

(P1) If $\psi:\mathbb{N}\to\mathbb{N}$ is strictly increasing, then $k\le\psi(k)$ for every $k\in\mathbb{N}$. Indeed $1\le\psi(1)$ because every natural number satisfies $1\le\cdot$ by claim 4 of Properties of the Order on the Natural Numbers; and if $k\le\psi(k)$ then $\psi(k)<\psi(k+1)$, so $k<\psi(k+1)$ and hence $k+1\le\psi(k+1)$.

(P2) If a sequence $(a_{m})$ in a metric space $(Y,d)$ converges to $a$ and $\psi:\mathbb{N}\to\mathbb{N}$ is strictly increasing, then $(a_{\psi(k)})_{k\in\mathbb{N}}$ converges to $a$. Indeed, given a positive $\varepsilon$, take $N$ as in Convergent Sequence in a Metric Space; for $k\ge N$ we have $\psi(k)\ge k\ge N$ by (P1), so $d(a_{\psi(k)},a)<\varepsilon$.

(P3) A composite of strictly increasing maps $\mathbb{N}\to\mathbb{N}$ is strictly increasing, directly from the definition.

We also use that $\lVert e_{j}\rVert=1$ for every $j\in[n]$, as recorded in Euclidean Space and Real Matrices: Standing Notation (vi).

Claim 1. Put $R'=R+1$. Since $0\le\lVert X_{1}\rVert\le R$ by claim 1 of Properties of the Norm of a Symmetric Real Matrix, the number $R'$ is positive, and $\lVert X_{m}\rVert\le R'$ for every $m$. By Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix (iii), for all $m\in\mathbb{N}$ and $j\in[n]$,

$$\lVert X_{m}e_{j}\rVert\le\lVert X_{m}\rVert\,\lVert e_{j}\rVert\le R' .$$

Hence every $X_{m}e_{j}$ lies in the closed ball $K=\bar{B}(0_{\mathbb{R}^{n}},R')$, which equals $\{\zeta\in\mathbb{R}^{n}:\lVert\zeta\rVert\le R'\}$ because $d_{E}(0_{\mathbb{R}^{n}},\zeta)=\lVert\zeta\rVert$ by claim 2 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square, and which is bounded by claim 2 of Elementary Properties of the Closed Ball in a Metric Space.

We show by induction on $l\in\{0,1,\dots,n\}$ that there is a strictly increasing $\varphi_{l}:\mathbb{N}\to\mathbb{N}$ such that for every $j\in[n]$ with $j\le l$ the sequence $\bigl(X_{\varphi_{l}(k)}e_{j}\bigr)_{k\in\mathbb{N}}$ converges in $(\mathbb{R}^{n},d_{E})$. For $l=0$ take $\varphi_{0}$ to be the identity map; the condition is vacuous. Suppose the assertion holds for some $l$ with $l<n$. The sequence $\bigl(X_{\varphi_{l}(k)}e_{l+1}\bigr)_{k\in\mathbb{N}}$ takes its values in the bounded set $K$, so by The Bolzano-Weierstrass Theorem in $\mathbb{R}^n$ there are a strictly increasing $\rho:\mathbb{N}\to\mathbb{N}$ and a point of $\mathbb{R}^{n}$ to which $\bigl(X_{\varphi_{l}(\rho(k))}e_{l+1}\bigr)_{k}$ converges. Put $\varphi_{l+1}=\varphi_{l}\circ\rho$, which is strictly increasing by (P3). For $j\le l$ the sequence $\bigl(X_{\varphi_{l+1}(k)}e_{j}\bigr)_{k}$ arises from the convergent sequence $\bigl(X_{\varphi_{l}(k)}e_{j}\bigr)_{k}$ by the strictly increasing map $\rho$, hence converges by (P2); and for $j=l+1$ it converges by construction. This completes the induction.

Put $\varphi=\varphi_{n}$ and, for $j\in[n]$, let $c_{j}\in\mathbb{R}^{n}$ be the limit of $\bigl(X_{\varphi(k)}e_{j}\bigr)_{k}$. By Convergence in Euclidean Space is Coordinatewise Convergence, for every $i\in[n]$ the sequence of real numbers $\bigl((X_{\varphi(k)}e_{j})_{i}\bigr)_{k}$ converges to $(c_{j})_{i}$. By Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix (i) and the fact, recorded in Orthonormal Families, Standard Basis Vectors, and Plane Rotations of Euclidean Space, that the $i$th coordinate of a point $z$ equals $z\cdot e_{i}$, we have $(X_{\varphi(k)})_{ij}=e_{i}\cdot(X_{\varphi(k)}e_{j})=(X_{\varphi(k)}e_{j})_{i}$. Hence for all $i,j\in[n]$ the real sequence $\bigl((X_{\varphi(k)})_{ij}\bigr)_{k}$ converges to $(c_{j})_{i}$.

Let $X$ be the real $n\times n$ matrix with $X_{ij}=(c_{j})_{i}$. For all $i,j$ the two real sequences $\bigl((X_{\varphi(k)})_{ij}\bigr)_{k}$ and $\bigl((X_{\varphi(k)})_{ji}\bigr)_{k}$ are equal term by term, because each $X_{m}$ is symmetric; by uniqueness of limits, Uniqueness of Limits in a Metric Space applied in the metric space of The Absolute Value Metric on the Real Numbers, we get $X_{ij}=X_{ji}$. Thus $X$ is symmetric and $X\in\mathcal{S}(n)$.

It remains to prove that $\bigl(X_{\varphi(k)}\bigr)_{k}$ converges to $X$ in $\bigl(\mathcal{S}(n),d_{\mathcal{S}(n)}\bigr)$. Put $\Sigma=\sum_{i=1}^{n}\sum_{j=1}^{n}1$, a real number satisfying $1\le\Sigma$ by claim 6 of Properties of Finite Sums; in particular $\Sigma$ and $\Sigma+\Sigma$ are positive and have multiplicative inverses. Let $\varepsilon\in\mathbb{R}$ be positive and put $\varepsilon'=\varepsilon\,(\Sigma+\Sigma)^{-1}$, which is positive. For each pair $i,j\in[n]$ choose $N_{ij}\in\mathbb{N}$ such that $k\ge N_{ij}$ implies $\bigl|(X_{\varphi(k)})_{ij}-X_{ij}\bigr|<\varepsilon'$, and put $N=\sum_{i=1}^{n}\sum_{j=1}^{n}N_{ij}$, a natural number; since every $N_{ij}$ is nonnegative, claim 6 of Properties of Finite Sums gives $N_{ij}\le N$ for all $i,j$. Let $k\ge N$. By Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix (v), applied to the matrix $X_{\varphi(k)}-X$, which lies in $\mathcal{S}(n)$ by claim 1 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, and by the entrywise description of the difference of matrices,

$$d_{\mathcal{S}(n)}\bigl(X_{\varphi(k)},X\bigr)=\bigl\lVert X_{\varphi(k)}-X\bigr\rVert\le\sum_{i=1}^{n}\sum_{j=1}^{n}\bigl|(X_{\varphi(k)})_{ij}-X_{ij}\bigr| .$$

Each summand is at most $\varepsilon'$, so by claim 1 of Comparison and Absolute Value Bounds for Finite Sums of Real Numbers, applied to the inner and then the outer sum, and by claim 3 of Properties of Finite Sums applied twice to pull the constant $\varepsilon'$ out, the right-hand side is at most $\varepsilon'\,\Sigma$, which equals $\varepsilon\,(\Sigma+\Sigma)^{-1}\Sigma$ and is therefore smaller than $\varepsilon$. This is the required convergence.

Claim 2. Let $z\in\mathbb{R}^{n}$. By claim 1 of Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum and claim 5 of The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite,

$$z\cdot(X_{m}z)-z\cdot(Xz)=z\cdot\bigl((X_{m}-X)z\bigr),$$

and by claim 2 of Properties of the Norm of a Symmetric Real Matrix,

$$\bigl|z\cdot(X_{m}z)-z\cdot(Xz)\bigr|\le\lVert X_{m}-X\rVert\,\lVert z\rVert^{2}=d_{\mathcal{S}(n)}(X_{m},X)\,\lVert z\rVert^{2}.$$

If $z$ is the origin then $\lVert z\rVert^{2}=0$ by claim 3 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square and the two sides of the asserted convergence agree for every $m$. Otherwise $\lVert z\rVert$ is positive by claims 1 and 3 there, hence so is $\lVert z\rVert^{2}$ by claim 5 of Elementary Order Arithmetic in an Ordered Field, and it has a positive multiplicative inverse. Given a positive $\varepsilon$, choose $N$ with $d_{\mathcal{S}(n)}(X_{m},X)<\varepsilon\,(\lVert z\rVert^{2})^{-1}$ for $m\ge N$; then $\bigl|z\cdot(X_{m}z)-z\cdot(Xz)\bigr|<\varepsilon$ for such $m$.

Claim 3. Let $z\in\mathbb{R}^{n}$. By The Positive Semidefinite Ordering on Symmetric Matrices the hypothesis $X_{m}\preceq Y_{m}$ gives $z\cdot(X_{m}z)\le z\cdot(Y_{m}z)$ for every $m$. By claim 2 the real sequences $\bigl(z\cdot(X_{m}z)\bigr)_{m}$ and $\bigl(z\cdot(Y_{m}z)\bigr)_{m}$ converge to $z\cdot(Xz)$ and $z\cdot(Yz)$ respectively, so claim 1 of Order Properties of Limits of Real Sequences gives $z\cdot(Xz)\le z\cdot(Yz)$. As $z$ was arbitrary, $X\preceq Y$ by The Positive Semidefinite Ordering on Symmetric Matrices.

Claim 4. Put $\lambda=a+\lVert C\rVert$, which is nonnegative because $0\le a$ and $0\le\lVert C\rVert$ by claim 1 of Properties of the Norm of a Symmetric Real Matrix. By claim 3 of Properties of the Norm of a Symmetric Real Matrix we have $C\preceq\lVert C\rVert I_{n}$. Moreover $\lambda I_{n}-\lVert C\rVert I_{n}=aI_{n}$, read entrywise from Scalar Multiple of a Real Matrix and Difference of Real Matrices, and $aI_{n}$ is positive semidefinite: by Vector, Entry and Comparison Bounds for the Norm of a Symmetric Real Matrix (vi) we have $z\cdot\bigl((aI_{n})z\bigr)=a\lVert z\rVert^{2}$ for every $z$, and this is nonnegative because $a$ is nonnegative by hypothesis and $\lVert z\rVert^{2}$ is nonnegative by claim 1 of Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square. Hence $\lVert C\rVert I_{n}\preceq\lambda I_{n}$ by claim 5 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure. Likewise $-aI_{n}-(-\lambda I_{n})=\lVert C\rVert I_{n}$, which is positive semidefinite by the same argument applied to the nonnegative number $\lVert C\rVert$, so $-\lambda I_{n}\preceq-aI_{n}$. Two applications of transitivity, claim 2 of The Positive Semidefinite Ordering is a Partial Order Compatible with the Linear Structure, now give

$$-\lambda I_{n}\preceq X\qquad\text{and}\qquad X\preceq\lambda I_{n},$$

and claim 3 of Properties of the Norm of a Symmetric Real Matrix yields $\lVert X\rVert\le\lambda=a+\lVert C\rVert$.