In linear algebra, a positivedefinite matrix is a matrix which in many ways is analogous to a positive real number. The notion is closely related to a positivedefinite symmetric bilinear form (or a sesquilinear form in the complex case).
The proper definition of positivedefinite is unambiguous for Hermitian matrices, but there is no agreement in the literature on how this should be extended for nonHermitian matrices, if at all. (See the section NonHermitian matrices below.)
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Definition
An n × n real symmetric matrix M is positive definite if z^{T}Mz > 0 for all nonzero vectors z with real entries (), where z^{T} denotes the transpose of z.
An n × n complex Hermitian matrix M is positive definite if z^{*}Mz > 0 for all nonzero complex vectors z, where z^{*} denotes the conjugate transpose of z. The quantity z^{*}Mz is always real because M is a Hermitian matrix.
Examples
The matrix is positive definite. For a vector with entries the quadratic form is when the entries z_{0}, z_{1} are real and at least one of them nonzero, this is positive.
The matrix is not positive definite. When the quadratic form at z is then
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