Vector- and Matrix-Valued Functions on a Compact Interval: Standing Conventions

(i) This setting extends Single-Variable Calculus on an Interval and Euclidean Space and Real Matrices: Standing Notation: their conventions and notation are in force. For $a,b\in\mathbb{R}$ with $a<b$, $[a,b]$ is an interval containing the two distinct points $a$ and $b$, by Basic Facts about Intervals of the Real Line (v).

(ii) The word "function" is used for "map" of the extended settings: a function $f:X\to Y$ is a map from $X$ to $Y$. A function given by an expression in its arguments, such as $t\mapsto A(t)^{\top}$, $r\mapsto\Phi(t,r)$, $(t,P)\mapsto A(t)P+PA(t)^{\top}+C(t)$, $t\mapsto R(t)^{-1}$ (where every $R(t)$ is invertible) or $u\mapsto x(a+b-u)$, exists and is unique, with the stated values; this is background, by Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (v), Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (iv), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (i), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (ii), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iii), Maps Formed Pointwise: Constant Maps, Pairs of Values, Maps Applied to Values, and Families of Maps (iv), Maps Formed Pointwise from Vector and Matrix Operations (i), Maps Formed Pointwise from Vector and Matrix Operations (ii), Projection, Evaluation and Currying Maps (i), Maps Given by Integrals of Vector- and Matrix-Valued Maps (i), Maps Given by Integrals of Vector- and Matrix-Valued Maps (ii) and The Reflection of a Closed Interval (i); further properties of the reflection $u\mapsto a+b-u$ of $[a,b]$ are The Reflection of a Closed Interval.

(iii) Unless stated otherwise, continuity of a function with values in $\mathbb{R}$, $\mathbb{R}^{q}$ or $\mathcal{M}_{p\times q}(\mathbb{R})$ ($p,q\in\mathbb{N}$) on a subset of its domain is understood as continuity on that subset for the following metrics: $d_{\mathbb{R}}$ on subsets of $\mathbb{R}$, $d_{E}$ on $\mathbb{R}^{q}$ and $d_{p\times q}$ on $\mathcal{M}_{p\times q}(\mathbb{R})$, as in the extended settings, and, on subsets of $\mathbb{R}\times\mathbb{R}^{q}$ and of $\mathbb{R}\times\mathcal{M}_{p\times q}(\mathbb{R})$, the product metric of $d_{\mathbb{R}}$ with $d_{E}$, respectively with $d_{p\times q}$, a metric by The Product Metric is a Metric (i). Background: Continuity of Vector- and Matrix-Valued Maps: the Componentwise Criterion, Norms and Algebraic Operations, Continuity of Projections, Negatives and Differences, and Derivatives on Subintervals and Continuity and Derivative of the Inverse of a Matrix-Valued Map.

(iv) For $a<b$ and $p,q\in\mathbb{N}$, differentiability, derivatives and derivative functions of functions from $[a,b]$ into $\mathbb{R}^{q}$ or $\mathcal{M}_{p\times q}(\mathbb{R})$ are as in Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (i), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (ii), Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iii) and Derivative of a Vector-Valued or Matrix-Valued Map on an Interval (iv), taken relative to $[a,b]$, and "differentiable", said of such a function, means differentiable on $[a,b]$; their Riemann integrability and integrals are as in Riemann Integral of a Vector-Valued or Matrix-Valued Map (i), Riemann Integral of a Vector-Valued or Matrix-Valued Map (ii), Riemann Integral of a Vector-Valued or Matrix-Valued Map (iii) and Riemann Integral of a Vector-Valued or Matrix-Valued Map (iv). Background: Rules for Derivatives of Vector- and Matrix-Valued Maps, Rules for the Riemann Integral of Vector- and Matrix-Valued Maps, The Fundamental Theorem of Calculus and the Mean Value Inequality for Vector- and Matrix-Valued Maps, Shifted Power Maps, and Integrating a Power Bound on the Derivative of a Matrix-Valued Map and Integrals Preserve Order, Symmetry and Positive Semidefiniteness.

(v) Background for computing with vectors and matrices: Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (i), Additive Laws for Vectors in $\mathbb{R}^q$ and for Real $p\times q$ Matrices: Group Laws and Rules for Differences (ii) and Products of Real Matrices with Zero Matrices, Negatives and Differences.