# Quadratic form

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The function, $LaTeX: x^TQx,$ where $LaTeX: x$ is an n-vector and $LaTeX: Q$ is an $LaTeX: \mbox{n} \times \mbox{n}$ matrix. Its gradient is $LaTeX: x^T[Q+Q^T].$ Typically, $LaTeX: Q$ is presumed symmetric, in which case its gradient is $LaTeX: 2x^TQ,$ and its eigenvalues are real.

• The quadratic form is:
• positive semi-definite if $LaTeX: x^T Qx \ge 0$ for all $LaTeX: x,$ in which case its eigenvalues are non-negative, and the quadratic form is a convex function.
• positive definite if $LaTeX: x^TQx > 0$ for all nonzero $LaTeX: x,$ in which case its eigenvalues are positive, and the quadratic form is a strongly convex function.
• negative semi-definite if $LaTeX: x^TQx \le 0$ for all $LaTeX: x,$ in which case its eigenvalues are non-positive, and the quadratic form is a concave function.
• negative definite if $LaTeX: x^TQx < 0$ for all nonzero $LaTeX: x,$ in which case its eigenvalues are negative, and the quadratic form is a strongly concave function.
• indefinite if it is none of the above.