Positive Definite Matrix
A Positive Definite Matrix is a symmetric matrix, [math]\displaystyle{ M }[/math] whose eigenvalues are all positive.
- Context:
- It can be mathematically stated as: A matrix [math]\displaystyle{ A }[/math] is said to be positive definite if [math]\displaystyle{ \overline{X}^TAX\gt 0 }[/math] for any complex vector [math]\displaystyle{ X \neq 0 }[/math] and [math]\displaystyle{ \overline{X}^TAX=0 }[/math], if and only if [math]\displaystyle{ X=0 }[/math].
- It can have all leading minors value positive.
- If a matrix [math]\displaystyle{ -A }[/math] is positive definite then the matrix [math]\displaystyle{ A }[/math] is called negative definite.
- It must, when multiplied with a non-zero vector (and its transpose), result in a Positive Real-Number.
- Example(s):
- [math]\displaystyle{ \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix} }[/math] is a positive definite matrix with all positive eigenvalues 5 and 2.
- [math]\displaystyle{ \begin{bmatrix} 1 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & 3 \end{bmatrix} }[/math] is a positive definite matrix with all positive eigenvalues 1,2 and 2.
- For a real square matrix [math]\displaystyle{ A }[/math], [math]\displaystyle{ A^TA }[/math] is a positive definite matrix.
- …
- Counter-Example(s):
- See: Kernel Function, Hessian Matrix.
References
2013
- (Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Positive-definite_matrix
- In linear algebra, a symmetric n × n real matrix M is said to be 'positive definite if zTMz is positive, for any non-zero column vector z of n real numbers; where zT denotes the transpose of z.
More generally, an n × n complex matrix M is said to be positive definite if z*Mz is real and positive for all non-zero complex vectors z ; where z* denotes the conjugate transpose of z. This property implies that M is an Hermitian matrix.
The negative definite, positive semi-definite, and negative semi-definite matrices are defined in the same way, except that the formula zTMz or z*Mz is required to be always negative, non-negative, and non-positive, respectively.
Positive definite matrices are closely related to a positive-definite symmetric bilinear forms (or sesquilinear forms in the complex case), and to inner products of vector spaces.
Some authors use more general definitions of "positive definite" that include some non-symmetric real matrices, or non-Hermitian complex ones. However, some of those extended definitions are incompatible, and may not be suitable for certain applications.
- In linear algebra, a symmetric n × n real matrix M is said to be 'positive definite if zTMz is positive, for any non-zero column vector z of n real numbers; where zT denotes the transpose of z.
2011
- (Weisstein - Positive Definite Matrix, 2011-Jul-17) ⇒ Eric W. Weisstein. (2011). “Hessian." From MathWorld -- A Wolfram Web Resource.
- QUOTE: An [math]\displaystyle{ n×n }[/math] complex matrix [math]\displaystyle{ A }[/math] is called positive definite if [math]\displaystyle{ R[x^*Ax]\gt 0 }[/math] (1) for all nonzero complex vectors [math]\displaystyle{ x }[/math] in [math]\displaystyle{ C^n }[/math], where [math]\displaystyle{ x^* }[/math] denotes the conjugate transpose of the vector [math]\displaystyle{ x }[/math]. In the case of a real matrix [math]\displaystyle{ A }[/math], equation (1) reduces to [math]\displaystyle{ x^(T)Ax\gt 0, }[/math] (2) where [math]\displaystyle{ x^(T) }[/math] denotes the transpose.