The Ordered Eigenvalues of a Symmetric Matrix Are Determined by the Matrix

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$, let $A\in\mathcal{S}(n)$ be a symmetric matrix, let $Q,P\in\mathrm{O}(n)$ be orthogonal matrices, and let $\lambda,\mu\in\mathbb{R}^{n}$.

(i) Uniqueness. If

$$A=(Q\operatorname{diag}(\lambda))Q^{\top}=(P\operatorname{diag}(\mu))P^{\top},\qquad\lambda_{1}\ge\dots\ge\lambda_{n},\qquad\mu_{1}\ge\dots\ge\mu_{n},$$

then $\lambda=\mu$.