In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$. We propose to show that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. For this, we denote by $X$ and $Y$ the matrices of $\mathcal{M}_n(\mathbb{R})$ such that $U = X + \mathrm{i}Y$. a) Show that there exists $\mu \in \mathbb{R}$ such that $X + \mu Y \in \mathrm{GL}_n(\mathbb{R})$. b) Show that $AX = XB$ and $AY = YB$. c) Conclude.
In this part, $A$ and $B$ denote two matrices of $\mathcal{M}_n(\mathbb{R})$. We assume that there exists a square matrix $U$ of size $n$, invertible, with complex coefficients, such that $U {}^t\bar{U} = I_n$ and $A = UBU^{-1}$, where $\bar{U}$ denotes the matrix whose coefficients are the conjugates of those of $U$.
We propose to show that there exists a matrix $P \in \mathrm{GL}_n(\mathbb{R})$ such that $A = PBP^{-1}$ and ${}^t A = P {}^t B P^{-1}$. For this, we denote by $X$ and $Y$ the matrices of $\mathcal{M}_n(\mathbb{R})$ such that $U = X + \mathrm{i}Y$.
a) Show that there exists $\mu \in \mathbb{R}$ such that $X + \mu Y \in \mathrm{GL}_n(\mathbb{R})$.
b) Show that $AX = XB$ and $AY = YB$.
c) Conclude.