QI.E
Matrices
Linear Transformation and Endomorphism Properties
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We consider the matrix with real coefficients $C \in \mathscr { M } _ { 7 } ( \mathbb { R } )$
$$C = \left( \begin{array} { l l l l l l l }
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
\mathbf { 1 } & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & \mathbf { 1 } & 0 & 0 & \mathbf { 1 } & 0 & 0 \\
0 & 0 & \mathbf { 1 } & \mathbf { 1 } & 0 & 0 & 0
\end{array} \right)$$
and $c$ the endomorphism of $\mathbb{R}^7$ whose matrix in the canonical basis is $C$.
Notation: if $f$ is a function of class $C^1$ from an open set $\mathscr{U}$ of $\mathbb{R}^d$ ($d \geqslant 1$) to $\mathbb{R}$, we denote, for every integer $i$ such that $1 \leqslant i \leqslant d$, $\partial_i f$ the partial derivative of $f$ with respect to its $i$-th variable.
In this section, we propose to study functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ that satisfy the condition $f \circ c = f$, that is, such that $$f\left(x_3 + x_4, x_2 + x_5, x_1, x_1, x_1, x_2 + x_5, x_3 + x_4\right) = f\left(x_1, x_2, x_3, x_4, x_5, x_6, x_7\right)$$ for all $(x_1, x_2, x_3, x_4, x_5, x_6, x_7) \in \mathbb{R}^7$.
I.E.1) What structure does the set $\mathscr{S}$ of functions $f$ of class $C^1$ from $\mathbb{R}^7$ to $\mathbb{R}$ such that $f \circ c = f$ possess? I.E.2) Show that such a function satisfies $f \circ c^n = f$ for every integer $n \geqslant 1$. I.E.3) Let $f \in \mathscr{S}$. Calculate the Jacobian matrix of $f \circ c$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Deduce a system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$. I.E.4) For $f \in \mathscr{S}$, calculate the Jacobian matrix of $f \circ c^2$ at $X = (x_1, x_2, x_3, x_4, x_5, x_6, x_7)$. Complete the system of equations relating the partial derivatives $\partial_1 f(X), \ldots, \partial_7 f(X)$ of $f$ at a point $X$ of $\mathbb{R}^7$ obtained in the previous question. I.E.5) Application: without further calculation, determine the linear forms $f$ on $\mathbb{R}^7$ that belong to $\mathscr{S}$.