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.


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