Positive Semi-Definite Matrix
A Positive Semi-Definite Matrix is a Hermitian matrix whose eigenvalues are nonnegative.
- AKA: Semipositive-Definite Matrix.
- 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 Matrix.
- Example(s):
- [math]\displaystyle{ \begin{bmatrix} 2 & 2 & 2 \\ 2 & 6 & 2 \\ 2 & 2 & 2 \end{bmatrix} }[/math] is a positive semi-definite matrix with all nonnegative eigenvalues 8, 2 and 0.
- a Kernel Matrix.
- …
- Counter-Example(s):
- See: Covariance Matrix, Positive Eigenvalued Matrix, Positive Matrix, Semidefinite Program (SDP), Definite Bilinear Form.
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.
- 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.
2012
- http://mathworld.wolfram.com/PositiveSemidefiniteMatrix.html
- QUOTE: A positive semidefinite matrix is a Hermitian matrix all of whose eigenvalues are nonnegative.
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.
- 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.