To see that let us focus on the case where and Q is positive definite.

This only works if the new matrix is still positive definite.

To show that the result is positive definite requires further proof.

In the general case, where A is not positive definite, we have

However the last condition alone is not sufficient for M to be positive definite.

Such a process exists because the given covariance is positive definite.

If we additionally have that the function is positive definite, i.e.

A function with this property is called positive definite.

The latter regularization is done on square of because may not be positive definite.

I start with a seed covariance matrix which is positive definite.

The second law of thermodynamics requires that the matrix be positive definite.

Under certain conditions on , the reduced matrix will be positive definite.

If A is symmetric and positive definite, then we even have

For this reason, positive definite matrices play an important role in optimization problems.

It takes the form of an equivalent statement, to the effect that a certain generalized function is positive definite.

The inner product of an element with itself is positive definite:

It admits a conserved quantity, but this is not positive definite.

Note that, in the above formulation, A and B need not be positive definite.

In fact, the matrix A is neither diagonally dominant nor positive definite.

Since all energies are measured relative to the vacuum, H is positive definite.

A Jordan triple system is said to be positive definite (resp.

The notation means that the matrix is positive definite.

The following properties are equivalent to M being positive definite:

This space has a positive definite form, making it a true Hilbert space.

The following matrix is not positive definite since .

If is a positive-definite matrix, for any real number :

Conditions 1 and 2 together define a positive-definite function.

In mathematics, the term positive-definite function may refer to a couple of different concepts.

The theory of zonal functions that are not necessarily positive-definite.

A positive matrix is not the same as a positive-definite matrix.

The expression under the square root is always positive if A is real and positive-definite.

In contrast to the positive-definite case, these vectors need not be linearly independent.

Gegenbauer polynomials also appear in the theory of Positive-definite functions.

It can be viewed as a particular type of positive-definite kernel where the underlying set has the additional group structure.

On the other hand, consider now a positive-definite function F on G.

Conversely, given a positive-definite function, one can define a unitary representation of G in a natural way.

A positive-definite matrix is a matrix with special properties.

Suppose, for simplicity, that the inner product on 3-space has positive-definite signature:

Conversely, every operator-valued positive-definite function arises in this way.

In 1947, Ostrowski proved that if is symmetric and positive-definite then for .

So, an inner product on a real vector space is a positive-definite symmetric bilinear form.

A symmetric matrix is positive-definite if and only if all its eigenvalues are positive.

This proves that the matrix is positive-definite.

(Strong ellipticity would furthermore require the symbol to be positive-definite.)

A mixed state, in this case, is a matrix that is Hermitian, positive-definite, and has trace 1.

Also unless otherwise stated, all examples have a Euclidean metric and so a positive-definite quadratic form.

The variance of X is a kxk symmetric positive-definite matrix V.

Thus, can be defined as the linear span of the set of continuous positive-definite functions on .

If A is again a symmetric positive-definite matrix, then (assuming all are column vectors)

The particle is a real excitation, meaning that states containing this particle have a positive-definite norm.