Let $M \in S _ { n + 1 } ( \mathbb { R } )$ be a symmetric matrix. We write $M$ in block form $$M = \left[ \begin{array} { l l } A & y \\ { } ^ { t } y & a \end{array} \right]$$ with $a \in \mathbb { R } , y \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $A \in S _ { n } ( \mathbb { R } )$. (a) If the spectrum of $A$ is $\operatorname { Sp } ( A ) = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right)$, show that there exist $U \in O _ { n + 1 } ( \mathbb { R } )$ and $z \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ such that $$U M ^ { t } U = \left[ \begin{array} { c c } \Delta \left( \mu _ { 1 } , \ldots , \mu _ { n } \right) & z \\ t _ { z } & a \end{array} \right]$$ (b) Deduce that there exist non-negative real numbers $\alpha _ { j }$ (for $j = 1 , \ldots , n$ ) such that $$\chi _ { M } = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } , \quad \text { where } \quad Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) .$$ (c) Show that $\operatorname { Sp } ( M )$ and $\operatorname { Sp } ( A )$ are interlaced.
Let $M \in S _ { n + 1 } ( \mathbb { R } )$ be a symmetric matrix. We write $M$ in block form
$$M = \left[ \begin{array} { l l } A & y \\ { } ^ { t } y & a \end{array} \right]$$
with $a \in \mathbb { R } , y \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ and $A \in S _ { n } ( \mathbb { R } )$.\\
(a) If the spectrum of $A$ is $\operatorname { Sp } ( A ) = \left( \mu _ { 1 } \geqslant \cdots \geqslant \mu _ { n } \right)$, show that there exist $U \in O _ { n + 1 } ( \mathbb { R } )$ and $z \in \mathcal { M } _ { n , 1 } ( \mathbb { R } )$ such that
$$U M ^ { t } U = \left[ \begin{array} { c c } \Delta \left( \mu _ { 1 } , \ldots , \mu _ { n } \right) & z \\ t _ { z } & a \end{array} \right]$$
(b) Deduce that there exist non-negative real numbers $\alpha _ { j }$ (for $j = 1 , \ldots , n$ ) such that
$$\chi _ { M } = ( X - a ) Q _ { 0 } - \sum _ { j = 1 } ^ { n } \alpha _ { j } \frac { Q _ { 0 } } { \left( X - \mu _ { j } \right) } , \quad \text { where } \quad Q _ { 0 } = \prod _ { k = 1 } ^ { n } \left( X - \mu _ { k } \right) .$$
(c) Show that $\operatorname { Sp } ( M )$ and $\operatorname { Sp } ( A )$ are interlaced.