Palindromic matrix polynomials, matrix functions and integral representations

We study the properties of palindromic quadratic matrix polynomials $\varphi(z)=P+Qz+Pz^2$, i.e., quadratic polynomials where the coefficients $P$ and $Q$ are square matrices, and where the constant and the leading coefficients are equal. We show that, for suitable choices of the matrix coefficients $P$ and $Q$, it is possible to characterize by means of $\varphi(z)$ well known matrix functions, namely the matrix square root, the matrix polar factor, the matrix sign and the geometric mean of two matrices. Finally we provide some integral representations of these matrix functions.