Symmetric Matrices and Quadratic Forms — Cheat sheet

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Core rule

Real symmetric AA has A=QΛQTA=Q\Lambda Q^T. In eigen-coordinates, xTAx=∑λiyi2x^TAx=\sum\lambda_i y_i^2.

Watch for

Positive semidefinite allows zero directions. A stationary point is a strict quadratic minimum only with positive definite curvature.