Let $A \in S_n^{++}(\mathbf{R})$ and let $g : t \in \mathbf{R} \mapsto \operatorname{det}(I_n + tA)$. Express, for all $t \in \mathbf{R}$, $g(t)$ using the eigenvalues of $A$. Deduce that $g$ is of class $C^\infty$ on $\mathbf{R}$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $g : t \in \mathbf { R } \mapsto \operatorname { det } \left( I _ { n } + t A \right)$. Express, for all $t \in \mathbf { R } , g ( t )$ using the eigenvalues of $A$. Deduce that $g$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

For $p \in \mathbb{K}[X]$, express $Jp$ in terms of the derivatives $p^{(k)}$ ($k \in \mathbb{N}$) of $p$, where $J$ is defined by $Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$.

Let $A \in S_n^{++}(\mathbf{R})$ and let $f : t \mapsto \ln(\operatorname{det}(I_n + tA))$. Show that $$\forall t \in \mathbf{R}_+, \quad \ln(\operatorname{det}(I_n + tA)) \leq \operatorname{Tr}(A) t.$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and let $f : t \mapsto \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right)$. Show that
$$\forall t \in \mathbf { R } _ { + } , \quad \ln \left( \operatorname { det } \left( I _ { n } + t A \right) \right) \leq \operatorname { Tr } ( A ) t$$

Prove that the endomorphism $D - I$ is invertible and express $L$ in terms of $(D-I)^{-1}$, where $L$ is defined by $Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A$ is of class $C^\infty$ on $\mathbf{R}$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A }$ is of class $C ^ { \infty }$ on $\mathbf { R }$.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$. We recall that the degree of the zero polynomial is by convention equal to $-1$.
Show that there exists a natural number $n(T)$ such that, for every polynomial $p \in \mathbb{K}[X]$, $$\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that there exists $\varepsilon _ { 0 } > 0$ such that, for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$, and let $n(T)$ be the natural number such that $\deg(Tp) = \max\{-1, \deg(p) - n(T)\}$ for every $p \in \mathbb{K}[X]$.
Deduce $\ker(T)$ in terms of $n(T)$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Show that $f_A(t) \underset{t \rightarrow 0}{=} \operatorname{det}(A) + \operatorname{det}(A) \operatorname{Tr}(A^{-1}M) t + o(t)$.
Hint: You may begin by treating the case where $A = I_n$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
Show that $f _ { A } ( t ) \underset { t \rightarrow 0 } { = } \operatorname { det } ( A ) + \operatorname { det } ( A ) \operatorname { Tr } \left( A ^ { - 1 } M \right) t + o ( t )$.
Hint: You may begin by treating the case where $A = I _ { n }$.

Let $T$ be a non-zero shift-invariant endomorphism of $\mathbb{K}[X]$.
Show that the following three assertions are equivalent:

1. [(1)] $T$ is invertible;
2. [(2)] $T1 \neq 0$;
3. [(3)] $\forall p \in \mathbb{K}[X], \deg(Tp) = \deg(p)$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Determine $f_A'(t)$ for all $t \in ]-\varepsilon_0, \varepsilon_0[$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Determine $f _ { A } ^ { \prime } ( t )$ for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. We admit that the function $\Phi : t \mapsto (A + tM)^{-1}$ is of class $C^1$ on $]-\varepsilon_0, \varepsilon_0[$. By noting that $\Phi(t) \times (A + tM) = I_n$, show that $$\Phi(t) \underset{t \rightarrow 0}{=} A^{-1} - A^{-1}MA^{-1} t + o(t).$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. We admit that the function $\Phi : t \mapsto ( A + t M ) ^ { - 1 }$ is of class $C ^ { 1 }$ on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$. By noting that $\Phi ( t ) \times ( A + t M ) = I _ { n }$, show that
$$\Phi ( t ) \underset { t \rightarrow 0 } { = } A ^ { - 1 } - A ^ { - 1 } M A ^ { - 1 } t + o ( t )$$

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$ Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. We define the application $\varphi _ { \alpha }$ by
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \right.$$
Show that $\varphi _ { \alpha }$ is differentiable on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$ and that
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \quad \varphi _ { \alpha } ^ { \prime } ( t ) = - \operatorname { Tr } \left( ( A + t M ) ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A + t M ) . \right.$$

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
Show that there exists a unique shift-invariant and invertible endomorphism $U$ such that $T = D \circ U$. Specify $U$ in the case $T = D$, then in the case $T = L$.

Let $T$ be a delta endomorphism of $\mathbb{K}[X]$.
For every polynomial $p \in \mathbb{K}[X]$ non-zero, verify that $\deg(Tp) = \deg(p) - 1$. Deduce $\ker(T)$ and the spectrum of $T$.

Let $Q$ be a delta endomorphism with associated polynomial sequence $(q_n)_{n \in \mathbb{N}}$.
Show that, for every natural number $n$, $$\forall (x,y) \in \mathbb{K}^2, \quad q_n(x+y) = \sum_{k=0}^n q_k(x) q_{n-k}(y)$$

Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number.
Show that the family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.

For $Q = D$, verify that $$\forall n \in \mathbb{N}, \quad q_n = \frac{X^n}{n!}$$