Proof of Continuity and Derivative of the Inverse of a Matrix-Valued Map

Write $B=A^{-1}$, so that $B(u)=A(u)^{-1}$ and, by Inverse Matrix and Invertible Real Square Matrix (i) and Inverse Matrix and Invertible Real Square Matrix (ii), $A(u)B(u)=I_{p}=B(u)A(u)$ for every $u\in I$. Distances on $I$ are $d_{\mathbb{R}}(u,v)=|u-v|$, by The Distance on the Real Numbers (i). Throughout, Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations is applied with $(X,d)=(\mathbb{R},d_{\mathbb{R}})$ and $S=I$.

Proof of (i). Suppose $A$ is continuous at $t$, and let $\varepsilon>0$. By Perturbations of an Invertible Matrix: the Invertible Matrices Form an Open Set and Inversion Is Continuous (iii), applied to the invertible matrix $A(t)$, there is a real $\delta_{1}>0$ such that every $M\in\mathcal{M}_{p}(\mathbb{R})$ with $d_{p}(A(t),M)<\delta_{1}$ is invertible and satisfies $d_{p}(A(t)^{-1},M^{-1})<\varepsilon$. By Continuous Map Between Metric Spaces (i), there is then a real $\delta>0$ such that every $u\in I$ with $|t-u|<\delta$ satisfies $d_{p}(A(u),A(t))<\delta_{1}$. For such $u$, $d_{p}(A(t),A(u))<\delta_{1}$, as $d_{p}$ is a metric (The Real Matrices with the Operator-Norm Distance Form a Complete Metric Space with Entrywise Convergence (i), Metric Space (iv)), so $d_{p}(A(t)^{-1},A(u)^{-1})<\varepsilon$, and $d_{p}(B(u),B(t))=d_{p}(A(t)^{-1},A(u)^{-1})$ by the symmetry of $d_{p}$ again (Metric Space (iv)). Hence $B$ is continuous at $t$, by Continuous Map Between Metric Spaces (i). If $A$ is continuous on $I$, this applies at every point of $I$ in place of $t$, so $B$ is continuous on $I$ by Continuous Map Between Metric Spaces (ii). If $B$ is continuous at $t$, then every entry map $B_{ij}=(A^{-1})_{ij}$ is continuous at $t$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii) with $s_{0}=t$; if $B$ is continuous on $I$, then every entry map of $B$ is continuous on $I$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (v), applied to Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii).

Proof of (ii). Let $u\in I$. As $A(u)\in\mathcal{S}(p)$ is positive definite, Invertibility of Positive Definite Matrices (ii) and Invertibility of Positive Definite Matrices (iii), applied to $A(u)$, give $A(u)^{-1}\in\mathcal{S}(p)$ and that $A(u)^{-1}$ is positive definite.

Proof of (iii). Suppose $A$ is differentiable at $t$, and let $D=A'(t)$, $A_{0}=A(t)$ and $B_{0}=B(t)$. By Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), every entry map $A_{ij}$ ($i,j\in[p]$) is differentiable at $t$ and $D_{ij}=A_{ij}'(t)$. Below, Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations is applied with $s_{0}=t$.

Continuity of $A$ and $B$ at $t$. Each $A_{ij}$ is continuous at $t$ relative to $I$ by The Increment Form of the Derivative, and Differentiability Implies Continuity (ii), so $A$ is continuous at $t$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii), and $B$ is continuous at $t$ by (i).

The difference quotient of $A$. Let $R:I\to\mathcal{M}_{p}(\mathbb{R})$ be the map with $R(u)=(u-t)^{-1}\bigl(A(u)-A_{0}\bigr)$ for $u\neq t$ and $R(t)=D$ (Maps Defined by Cases (i)). By Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv), its entry maps satisfy $R_{ij}(u)=\bigl(A_{ij}(u)-A_{ij}(t)\bigr)/(u-t)$ for $u\neq t$ and $R_{ij}(t)=A_{ij}'(t)$. Each $R_{ij}$ is continuous at $t$: given $\varepsilon>0$, as $A_{ij}'(t)$ is a derivative of $A_{ij}$ at $t$ (Derivative of a Real Function on an Interval (ii)), Derivative of a Real Function on an Interval (i) gives a real $\delta>0$ such that every $u\in I$ with $0<|u-t|<\delta$ satisfies $|R_{ij}(u)-R_{ij}(t)|<\varepsilon$, and for $u=t$ the difference is $0<\varepsilon$; as $d_{\mathbb{R}}(y,z)=|y-z|$, this is continuity of $R_{ij}$ at $t$ relative to $I$ by Continuous Map Between Metric Spaces (i). By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii), $R$ is continuous at $t$.

The difference quotient of $B$. For $u\in I$, by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (ii), Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iii) and Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iv), a difference $M-N$ of matrices being $M+(-1)N$, and a negative $-M$ being $(-1)M$, by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv), Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field (ii) (entrywise) and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii),

$$B(u)\bigl(A(u)-A_{0}\bigr)B_{0}=\bigl(B(u)A(u)\bigr)B_{0}-B(u)\bigl(A_{0}B_{0}\bigr)=I_{p}B_{0}-B(u)I_{p}=B_{0}-B(u).$$

Since $B(u)-B_{0}=(-1)\bigl(B_{0}-B(u)\bigr)$, entrywise by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) and Rules of Arithmetic in a Commutative Ring: Zero, Signs and Squares, and No Zero Divisors in a Field (ii) and hence by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii), multiplying by $(u-t)^{-1}$ for $u\neq t$ and using Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (iii) and computing entrywise gives

$$(u-t)^{-1}\bigl(B(u)-B_{0}\bigr)=-B(u)R(u)B_{0}\qquad(u\in I,\ u\neq t).$$

Let $Q:I\to\mathcal{M}_{p}(\mathbb{R})$ be the map $u\mapsto-B(u)R(u)B_{0}=(-1)\bigl((B(u)R(u))B_{0}\bigr)$ (Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i) and Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), with the matrix product and the scalar multiple as the maps of The Matrix Product, Matrix-Vector Product, Transpose, Trace, Identity Matrix and Standard Basis Vectors Exist and Are Unique (i) and Pointwise Operations on Real-Valued Maps Exist and Are Unique (ii)). The constant maps $u\mapsto B_{0}$ and $u\mapsto-1$ are continuous at $t$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv) and Continuity of Sums and Products of Real-Valued Functions on a Metric Space (i), and $B$ and $R$ are continuous at $t$; so $Q$, obtained by two matrix products and a scalar multiple, is continuous at $t$ by Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (iv). Put $E=Q(t)$; as $B(t)=B_{0}$ and $R(t)=D$, $E=(-1)\bigl((B_{0}D)B_{0}\bigr)=-B_{0}DB_{0}$, the triple product not depending on the bracketing by Algebraic Rules for Matrix Products, Matrix-Vector Products and Transposes (i).

Reading off the entries. Let $i,j\in[p]$ and $\varepsilon>0$. By Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations (ii), the entry map $Q_{ij}$ is continuous at $t$, so, by Continuous Map Between Metric Spaces (i) with $d_{\mathbb{R}}(y,z)=|y-z|$, there is a real $\delta>0$ such that every $u\in I$ with $|u-t|<\delta$ satisfies $|Q_{ij}(u)-E_{ij}|<\varepsilon$, since $E_{ij}=Q(t)_{ij}=Q_{ij}(t)$ and $|t-u|=|u-t|$. For $u\in I$ with $0<|u-t|<\delta$, the displayed identity and Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iv) give $Q_{ij}(u)=\bigl(B_{ij}(u)-B_{ij}(t)\bigr)/(u-t)$, so

$$\Bigl|\frac{B_{ij}(u)-B_{ij}(t)}{u-t}-E_{ij}\Bigr|<\varepsilon .$$

Hence $E_{ij}$ is a derivative of the entry map $B_{ij}$ at $t$ (Derivative of a Real Function on an Interval (i)), so $B_{ij}$ is differentiable at $t$ with $B_{ij}'(t)=E_{ij}$ (Derivative of a Real Function on an Interval (ii)). As this holds for all $i,j\in[p]$, $B$ is differentiable at $t$ by Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), and $B'(t)$ and $E$ have the same entries, so $B'(t)=E=-A(t)^{-1}A'(t)A(t)^{-1}$ by Real Matrices: the Set of Real m x n Matrices with Their Entrywise Operations (iii).