Definite quadratic form

Definiteness is a term from the mathematical subfield of linear algebra. It describes the sign real quadratic forms can take that are generated by matrices or more generally by bilinear forms.

Definiteness of bilinear forms and Sesquilinearformen

It is a vector space over the real ( or complex ) numbers.

A symmetric bilinear form (or a Hermitian sesquilinear ) is called

Each for all, applies. Note that in the complex case, because of the required Hermiteizität is always real. If none of these conditions is called the indefinite form. It is in this case assumes both positive and negative values.

So, the above conditions imply that the associated quadratic form is positive definite, positive semidefinite, negative definite, negative semidefinite or is indefinite.

Occasionally these terms are introduced in the real case for arbitrary, not necessarily symmetric bilinear forms. ( In the complex case one would have to additionally require that for all the value is real. However It follows from this that the sesquilinear form is Hermitian. )

A positive definite symmetric bilinear form (or Hermitian sesquilinear ) is called scalar product. For example, the standard scalar product on the (or ) is positive definite.

Definiteness of matrices

Each square matrix describes a bilinear form on (or a sesquilinear form on ). This is called a square matrix therefore positive definite if this property applies to the plane defined by the matrix bilinear or sesquilinear form. Accordingly, one can also define other properties. This means: Any ( optionally symmetric or Hermitian ) matrix

For all -line column vectors, where the row vector that emerges from the column vector by transposing.

In the complex case of the vector on the left must be transposed to a row vector and additional complex-conjugate are ( Hermitian adjoint, rather than merely ). Thus the inequalities make sense, the left side for each possible must be real. This is exactly the case when the matrix is Hermitian.

A matrix which is neither positive nor negative semidefinite is called " indefinite ". If and only accepts (or ) both positive and negative values.

Criteria for definiteness

Eigenvalues

A square symmetric (or Hermitian ) matrix is then exactly

Therefore any method can be used for the determination or estimation of eigenvalues ​​to determine the definiteness of the matrix. One way are the Gerschgorin circuits that allow to estimate the spectrum of at least. This is often enough to determine the definiteness. The Gerschgorin circles give based on the entries of the matrix quantities in the complex plane to where the eigenvalues ​​are contained, in the case of symmetric matrices intervals on the real axis. Thus, it is sometimes possible to easily determine the definiteness of a matrix. For details, particularly on the signature of symmetric bilinear forms and matrices, see inertia Sylvester's theorem.

Principal minors

A symmetric or Hermitian matrix is positive definite if and only if all the leading principal minors of are positive. From the fact that is exactly then negative definite if it is positive definite, it follows: if and only negative definite if the signs of the leading principal minors alternate, that is, if all the odd leading principal minors and all even are negative positive.

• For semidefiniteness there is no criterion that takes into account only the leading principal minors. This can be seen already in the diagonal matrix with entries 0 and -1. However, the corresponding statement is true if all the principal minors, and not only the leading principal minors are non-negative.

• For non- Hermitian matrices, the criterion does not apply. An example of this is the indefinite matrix whose principal minors are all positive.

• The criterion is often called Sylvester criterion. Chance the name " Hurwitz criterion " is used, although this originally referred only to Hurwitz matrices.

Gaussian elimination method

A real symmetric square matrix is positive definite if the Gaussian elimination with diagonal strategy, that is, without row permutations, can be performed with n positive pivot elements. This condition is particularly suitable for cases where anyway the Gauss method must be applied.

Cholesky decomposition

A symmetric matrix is positive definite if and only if there is a Cholesky decomposition, with a regular lower triangular matrix.

Diagonal dominant matrices

If a matrix symmetric and strictly diagonally dominant and are all diagonal elements of positive, is positive definite.

The converse is not true. The matrix

Is positive definite, but not strictly diagonally dominant.

Symmetrical share of general matrices

A real square matrix that is not necessary symmetric, positive definite if and only if its symmetric part

Is positive definite. The same applies to " negative definite " and " positive" or " negative semidefinite " for.

For complex matrices A, the situation is completely different. One can consider the Hermitian part and the schiefhermiteschen share for every complex matrix A.

The matrix is ​​then Hermitian, and it is. is positive definite if the schiefhermitesche share is equal to 0 and the Hermitian component which coincides with consequently positive definite.

Sufficient criterion for positive semidefiniteness

For all matrices applies: is positive semidefinite.

Importance

• If the matrix is ​​symmetric ( Hermitian ) and positive definite, then a scalar product is defined by (respectively).
• The restriction of a positive definite bilinear or sesquilinear form on a subspace is again positive definite, ie, in particular non-degenerate. This fact allows the decomposition of a room in a subspace and its orthogonal complement.
• The definiteness of the Hessian matrix plays in the study of critical points of a function, so the extreme value calculation play a crucial role.