grandes-ecoles 2023 Q8

grandes-ecoles · France · mines-ponts-maths1__pc Matrices Matrix Decomposition and Factorization

Let $A \in S_n^{++}(\mathbf{R})$ and $B \in S_n(\mathbf{R})$. Show that there exists a diagonal matrix $D \in M_n(\mathbf{R})$ and $Q \in GL_n(\mathbf{R})$ such that $B = QDQ^\top$ and $A = QQ^\top$. What can be said about the diagonal elements of $D$ if $B \in S_n^{++}(\mathbf{R})$?
Hint: You may use question 3.

Let $A \in S_n^{++}(\mathbf{R})$ and $B \in S_n(\mathbf{R})$. Show that there exists a diagonal matrix $D \in M_n(\mathbf{R})$ and $Q \in GL_n(\mathbf{R})$ such that $B = QDQ^\top$ and $A = QQ^\top$. What can be said about the diagonal elements of $D$ if $B \in S_n^{++}(\mathbf{R})$?

Hint: You may use question 3.

Paper Questions

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