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2025-05 Commutativity and the matrix exponential

Let \( X \in \mathbb{R}^{n \times n} \) be a symmetric matrix with eigenvalues \( \lambda_i \) and orthonormal eigenvectors \( u_i \). The spectral decomposition gives \( X = \sum_{i=1}^n \lambda_i u_i u_i^\top \). For a function \( f : \mathbb{R} \to \mathbb{R} \), define \( f(X) := \sum_{i=1}^n f(\lambda_i) u_i u_i^\top \). Let \( X, Y \in \mathbb{R}^{n \times n} \) be symmetric. Is it always true that \( e^{X+Y} = e^X e^Y \)? If not, under what conditions does the equality hold?

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