positive-definite matrix

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

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

The function on the domain of positive-definite matrices is convex.

Then, the positive elements are the positive-definite matrices.

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

The lower triangular matrix is the 'Cholesky triangle' of the original, positive-definite matrix.

Gaussian elimination is numerically stable for diagonally dominant or positive-definite matrices.

Positive-definite kernel, a generalization of a positive-definite matrix.

In operator theory, a branch of mathematics, a positive definite kernel is a generalization of a positive-definite matrix.

Let M be a symmetric and N a symmetric and positive-definite matrix.

The diffusion tensor, a 3 x 3 symmetric positive-definite matrix, offers a straightforward solution to both of these goals.

P are all positive-definite matrices, the problem is convex and can be readily solved using interior point methods, as done with semidefinite programming.

Since is symmetric, positive-definite matrix, it follows that all eigenvalues of satisfy , ().

However, a positive-definite matrix has precisely one positive-definite square root, which can be called its principal square root.

A sufficient matrix is a generalization both of a positive-definite matrix and of a P-matrix, whose principal minors are each positive.

Mathematically this means that the result of subtracting the expected squared error (which is not usually known) from is a semidefinite or positive-definite matrix.

In statistics, a matrix gamma distribution is a generalization of the gamma distribution to positive-definite matrices.

The Wishart distribution is a multivariate generalization of the gamma distribution (samples are positive-definite matrices rather than positive real numbers).

A symmetric, and a symmetric and positive-definite matrix can be simultaneously diagonalized, although not necessarily via a similarity transformation.

The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices.

For a given pair of real symmetric positive-definite matrices, and a given non-zero vector , the generalized Rayleigh quotient is defined as:

The variance of X is a kxk symmetric positive-definite matrix V. The multivariate normal distribution is a special case of the elliptical distributions.

Where sums of squares appear in univariate analysis of variance, in multivariate analysis of variance certain positive-definite matrices appear.

(B is a Hermitian positive-definite matrix and u the conjugate transpose of u) the Pythagorean theorem is:

In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices.