Given is a matrix $\boldsymbol{A}\in\mathbb{R}^{n\times n}$ with linearly independent eigenvectors $\boldsymbol{u}_1,...,\boldsymbol{u}_n$ and corresponding eigenvalues $\lambda_1,...,\lambda_n$. Additionally is given a $\delta\in\mathbb{R}_+$.
Assuming that the eigenvalues are real, then is it universally true that $$(e^{\boldsymbol{A}\delta}-\boldsymbol{I})^T(e^{\boldsymbol{A}t}-\boldsymbol{I})$$ is positive semi-definite (PSD) for $t\in[0,\delta]$?
And if that is the case, then is it also true that $$(e^{\boldsymbol{A}\delta}-\boldsymbol{I})^T(e^{\boldsymbol{A}\delta}-\boldsymbol{I}) - (e^{\boldsymbol{A}\delta}-\boldsymbol{I})^T(e^{\boldsymbol{A}t}-\boldsymbol{I})$$ is also PSD for the same interval?
Thanks!