In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
symmetric matrix with
real entries is positive-definite if the real number
is positive for every nonzero real
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, ...
where
is the
transpose of More generally, a
Hermitian matrix (that is, a
complex matrix equal to its
conjugate transpose) is
positive-definite if the real number
is positive for every nonzero complex column vector
where
denotes the conjugate transpose of
Positive semi-definite matrices are defined similarly, except that the scalars
and
are required to be positive ''or zero'' (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.
A matrix is thus positive-definite if and only if it is the matrix of a
positive-definite quadratic form or
Hermitian form. In other words, a matrix is positive-definite if and only if it defines an
inner product.
Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix is positive-definite if and only if it satisfies any of the following equivalent conditions.
* is
congruent with a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ...
with