Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We denote by $0 < \lambda _ { 1 } < \cdots < \lambda _ { d }$ the $d$ eigenvalues of $A$ (distinct pairwise) and $F _ { 1 } , \ldots , F _ { d }$ the associated eigenspaces. We consider the linear map from $\mathbb { R } ^ { N }$ to $\mathbb { R } ^ { N }$: $$x \mapsto \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 1 / 2 } p _ { F _ { i } } ( x )$$ where $p _ { F _ { i } }$ is the orthogonal projection (for the canonical inner product) onto $F _ { i }$. We denote by $A ^ { 1 / 2 }$ the matrix associated with this linear map in the canonical basis. a) We write $A = U D U ^ { T }$, where $D \in \mathcal { M } _ { N } ( \mathbb { R } )$ is the diagonal matrix containing the eigenvalues of $A$ in increasing order, with their multiplicities, and $U$ an orthogonal matrix. We denote by $D ^ { 1 / 2 }$ the diagonal matrix whose diagonal coefficients are the square roots of those of $D$. Show that $A ^ { 1 / 2 } = U D ^ { 1 / 2 } U ^ { T }$. b) Show that $A ^ { 1 / 2 } \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, that $A ^ { 1 / 2 } A ^ { 1 / 2 } = A$, and that $A ^ { 1 / 2 }$ commutes with $A$. c) Show that, for all $x \in \mathbb { R } ^ { N } , \| x \| _ { A } = \left\| A ^ { 1 / 2 } x \right\|$, where $\| x \| _ { A }$ is the norm defined in question 4.
Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We denote by $0 < \lambda _ { 1 } < \cdots < \lambda _ { d }$ the $d$ eigenvalues of $A$ (distinct pairwise) and $F _ { 1 } , \ldots , F _ { d }$ the associated eigenspaces. We consider the linear map from $\mathbb { R } ^ { N }$ to $\mathbb { R } ^ { N }$:
$$x \mapsto \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 1 / 2 } p _ { F _ { i } } ( x )$$
where $p _ { F _ { i } }$ is the orthogonal projection (for the canonical inner product) onto $F _ { i }$. We denote by $A ^ { 1 / 2 }$ the matrix associated with this linear map in the canonical basis.
a) We write $A = U D U ^ { T }$, where $D \in \mathcal { M } _ { N } ( \mathbb { R } )$ is the diagonal matrix containing the eigenvalues of $A$ in increasing order, with their multiplicities, and $U$ an orthogonal matrix. We denote by $D ^ { 1 / 2 }$ the diagonal matrix whose diagonal coefficients are the square roots of those of $D$. Show that $A ^ { 1 / 2 } = U D ^ { 1 / 2 } U ^ { T }$.
b) Show that $A ^ { 1 / 2 } \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, that $A ^ { 1 / 2 } A ^ { 1 / 2 } = A$, and that $A ^ { 1 / 2 }$ commutes with $A$.
c) Show that, for all $x \in \mathbb { R } ^ { N } , \| x \| _ { A } = \left\| A ^ { 1 / 2 } x \right\|$, where $\| x \| _ { A }$ is the norm defined in question 4.