Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$. 


I want to know when this matrix is also an involution, i.e., $P^2 = I$. 


If there is any characterization of such $A$, please kindly share. Thank you. 



Positive definite matrices diagonalised by orthogonal matrices that are also involutions
Let $n \times n$ matrix $\rm A$ be symmetric and positive definite. Since $\rm A$ is symmetric, it is diagonalizable. Hence, there exists a (non-singular) matrix $\rm P$ such that $\mathrm A = \mathrm P \,\mbox{diag} (\lambda\_1, \lambda\_2, \dots, \lambda\_n) \,\mathrm P^{-1}$, where $\lambda\_1, \lambda\_2, \dots, \lambda\_n > 0$. Suppose $\rm P$ is orthogonal and involutory — and, thus, symmetric. Hence,


$$\mathrm A = \mathrm P \,\mbox{diag} (\lambda\_1, \lambda\_2, \dots, \lambda\_n) \,\mathrm P$$


Since $\rm P$ is symmetric and involutory, it has a spectral decomposition and its eigenvalues are $\pm 1$


$$\mathrm P = \mathrm V \,\mbox{diag} (\sigma\_1, \sigma\_2, \dots, \sigma\_n) \,\mathrm V^\top$$


where $\mathrm V$ is an orthogonal matrix and $\sigma\_1, \sigma\_2, \dots, \sigma\_n = \pm 1$. Thus, we have the parameterization


$$\mathrm A = \mathrm V \,\mbox{diag} (\sigma\_1, \sigma\_2, \dots, \sigma\_n) \,\mathrm V^\top \,\mbox{diag} (\lambda\_1, \lambda\_2, \dots, \lambda\_n) \,\mathrm V \,\mbox{diag} (\sigma\_1, \sigma\_2, \dots, \sigma\_n) \,\mathrm V^\top$$


