Let two matrices $A, B \in \mathcal{M}_n(\mathbf{R})$ such that there exists a matrix $P \in GL_n(\mathbf{R})$ satisfying $A = P^\top B P$. Show that $d(B) \geq d(A)$ then that $d(B) = d(A)$.
Let two matrices $A, B \in \mathcal{M}_n(\mathbf{R})$ such that there exists a matrix $P \in GL_n(\mathbf{R})$ satisfying $A = P^\top B P$. Show that $d(B) \geq d(A)$ then that $d(B) = d(A)$.