Positive Semi-Definite Matrix

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

• Context:
  • It can be mathematically stated as: A matrix [math]\displaystyle{ A }[/math] is said to be positive semi definite if [math]\displaystyle{ \overline{X}^TAX \geq 0 }[/math] for any complex vector [math]\displaystyle{ X }[/math].
  • It can have all leading minors value nonnegative.
  • If a matrix [math]\displaystyle{ -A }[/math] is positive semi-definite then the matrix [math]\displaystyle{ A }[/math] is called Negative Semi-Definite.
• Example(s):
  • [math]\displaystyle{ \begin{bmatrix} 0 & 1 & 1 \\ \sqrt{2} & 2 & 0 \\ 0 & 1 & 1 \end{bmatrix} }[/math] is a sositive semi-definite matrix with all nonnegative eigenvalues 2.79,0.21 and 0.
  • a Kernel Matrix.
• See: Negative Semi-Definite Matrix, Positive Definite Matrix, Covariance Matrix, Semipositive-Definite Matrix, Negative Definite Matrix, Negative Semidefinite Matrix, Positive Eigenvalued Matrix, Positive Matrix, Semidefinite Program (SDP).

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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

• http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html
  • QUOTE: A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative.

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