We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ Let $M = \left( m _ { i j } \right) \in S _ { n } ( \mathbb { R } )$. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n } \right) \in \mathbb { R } ^ { n }$. We propose to prove that, for all $s \in \{ 1 , \ldots , n \}$, we have: $$\sum _ { i = 1 } ^ { s } m _ { i i } \leqslant \sum _ { i = 1 } ^ { s } \lambda _ { i }$$ (a) What do you think of the case $s = n$ ? (b) Express $\sum _ { i = 1 } ^ { n - 1 } m _ { i i }$ in terms of the eigenvalues of the matrix $M _ { \leqslant n - 1 }$ obtained by removing the last row and last column of $M$. Deduce inequality (3) when $s = n - 1$. (c) By proceeding by induction on $n$, show inequality (3), for all $s \in \{ 1 , \ldots , n \}$.
We consider the application
$$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$
Let $M = \left( m _ { i j } \right) \in S _ { n } ( \mathbb { R } )$. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \cdots \geqslant \lambda _ { n } \right) \in \mathbb { R } ^ { n }$. We propose to prove that, for all $s \in \{ 1 , \ldots , n \}$, we have:
$$\sum _ { i = 1 } ^ { s } m _ { i i } \leqslant \sum _ { i = 1 } ^ { s } \lambda _ { i }$$
(a) What do you think of the case $s = n$ ?\\
(b) Express $\sum _ { i = 1 } ^ { n - 1 } m _ { i i }$ in terms of the eigenvalues of the matrix $M _ { \leqslant n - 1 }$ obtained by removing the last row and last column of $M$. Deduce inequality (3) when $s = n - 1$.\\
(c) By proceeding by induction on $n$, show inequality (3), for all $s \in \{ 1 , \ldots , n \}$.