Positive Semi-Definite Matrix

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A Positive Semi-Definite Matrix is a Hermitian matrix whose eigenvalues are nonnegative.



References

2013

  • http://en.wikipedia.org/wiki/Positive-definite_matrix#Positive-semidefinite
    • M is called semipositive-definite (or sometimes nonnegative-definite) if :[math]\displaystyle{ x^{*} M x \geq 0 }[/math] for all x in Cn (or, all x in Rn for the real matrix), where x* is the conjugate transpose of x.

      A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent.

      For any matrix A, the matrix A*A is positive semidefinite, and rank(A) = rank(A*A). Conversely, any Hermitian positive semidefinite matrix M can be written as M = A*A ; this is the Cholesky decomposition.

2012

2011

  • (Sammut & Webb, 2011) ⇒ Claude Sammut, and Geoffrey I. Webb. (2011). “Positive Semidefinite.” In: (Sammut & Webb, 2011) p.779
    • QUOTE: A symmetric m×m matrix K satisfying ∀x ∈ cm : x*Kx ≥ 0 is called positive semidefinite. If the equality only holds for x=0⃗ the matrix is positive definite.

      A function k : X ×X → c, X≠∅, is positive (semi-) definite if for all m ∈ n and all x1, …, xm ∈ X the m ×m matrix K⃗ with elements Kij : = k(xi, xj) is positive (semi-) definite.

      Sometimes the term strictly positive definite is used instead of positive definite, and positive definite refers then to positive semidefiniteness.