The Ordered Eigenvalues of a Symmetric Matrix

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$ and let $A\in\mathcal{S}(n)$ be a symmetric matrix.

(i) Ordered eigenvalues. $\lambda(A)=(\lambda_{1}(A),\dots,\lambda_{n}(A))$ is the non-increasing tuple $\lambda\in\mathbb{R}^{n}$ of any decomposition

$$A=(Q\operatorname{diag}(\lambda))Q^{\top},\qquad Q\in\mathrm{O}(n),\qquad\lambda_{1}\ge\dots\ge\lambda_{n}$$

(one exists by The Spectral Theorem for Real Symmetric Matrices (i), and $\lambda$ does not depend on it by The Ordered Eigenvalues of a Symmetric Matrix Are Determined by the Matrix (i)). Its components $\lambda_{1}(A)\ge\dots\ge\lambda_{n}(A)$ are the ordered eigenvalues of $A$.