In linear algebra, a positive-definite matrix is a 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).
The proper definition of positive-definite is unambiguous for Hermitian matrices, but there is no agreement in the literature on how this should be extended for non-Hermitian matrices, if at all. (See the section Non-Hermitian matrices below.)
An n × n real symmetric matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z.
An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z. The quantity z*Mz is always real because M is a Hermitian matrix.
The matrix is positive definite. For a vector with entries the quadratic form is when the entries z0, z1 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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