Positive-Definite Matrix
A Positive-Definite Matrix is a symmetric real-number matrix, [math]\displaystyle{ M }[/math] where [math]\displaystyle{ z^TMz }[/math] is positive for every non-zero column vector [math]\displaystyle{ z }[/math] of [math]\displaystyle{ n }[/math] real numbers.
- Context:
- It can be represented as a Hermitian Matrix such that ...
- Example(s):
- an Identity Matrix, such as [math]\displaystyle{ I = \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} }[/math]
- [math]\displaystyle{ M = \begin{bmatrix} 2&-1&0\\-1&2&-1\\0&-1&2 \end{bmatrix} }[/math]
- Counter-Example(s):
- See: Positive Real Number, Bilinear Form, Transpose, Conjugate Transpose, Definite Bilinear Form, Sesquilinear Form, Positive Matrix, Conjugate Gradient, Conjugate Gradient Method.
References
2015
- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Positive-definite_matrix Retrieved:2015-1-19.
- In linear algebra, a symmetric real matrix M is said to be positive definite if zTMz is positive for every non-zero column vector z of n real numbers. Here zT denotes the transpose of z.
More generally, an Hermitian matrix M is said to be positive definite if z*Mz is real and positive for all non-zero column vectors z of n complex numbers. Here z* denotes the conjugate transpose of z.
The negative definite, positive semi-definite, and negative semi-definite matrices are defined in the same way, except that the expression zTMz or z*Mz is required to be always negative, non-negative, and non-positive, respectively.
Positive definite matrices are closely related to positive-definite symmetric bilinear forms (or sesquilinear forms in the complex case), and to inner products of vector spaces. [1]
Some authors use more general definitions of "positive definite" that include some non-symmetric real matrices, or non-Hermitian complex ones.
- In linear algebra, a symmetric real matrix M is said to be positive definite if zTMz is positive for every non-zero column vector z of n real numbers. Here zT denotes the transpose of z.
2009
- http://en.wikipedia.org/wiki/Positive-definite_matrix
- In linear algebra, a positive-definite matrix is a (Hermitian) matrix which in many ways is analogous to a positive real number. The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).