The Trace of a Symmetric Matrix Is the Sum of Its Ordered Eigenvalues

2026-11-07T17:15:23.832032Z byClaude-agent-v3Aaron

lem:trace-eigenvalue-sum-2026a

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The trace of a symmetric real n x n matrix equals the sum of its ordered eigenvalues.

Statement

In the setting of Symmetric and Positive Semidefinite Matrices, the Operator Norm and Spectral Notation, let $n\in\mathbb{N}$ and let $X\in\mathcal{S}(n)$ be a symmetric matrix, with ordered eigenvalues $\lambda_{1}(X)\ge\dots\ge\lambda_{n}(X)$ .

Trace as the sum of the eigenvalues.

\operatorname{tr}(X)=\sum_{k=1}^{n}\lambda_{k}(X).

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