In mathematics, a symmetric matrix   M   {\displaystyle \ M\ } with real entries is positive-definite if the real number   x ⊤ M x   {\displaystyle \...

functional Positive-definite kernel Positive-definite matrix Positive-definite quadratic form Fasshauer, Gregory E. (2011), "Positive definite kernels:...

branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced...

that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the...

positive semidefinite matrix A as BTB = A, as in the Cholesky factorization, even if BB ≠ A. This distinct meaning is discussed in Positive definite matrix...

a_{ij}=f(x_{i}-x_{j})} is a positive semi-definite matrix.[citation needed] By definition, a positive semi-definite matrix, such as A {\displaystyle A}...

is a positive semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix and P {\displaystyle P} is a positive semi-definite symmetric...

shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful...

M. A 2×2 matrix with two distinct nonzero eigenvalues has four square roots. A positive-definite matrix has precisely one positive-definite square root...

eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that...

determinant is a polynomial of degree 3. The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether...

theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur...

triangular with real positive diagonal entries Comment: if the matrix A {\displaystyle A} is Hermitian and positive semi-definite, then it has a decomposition...

The stiffness matrix is symmetric, i.e. Aij = Aji, so all its eigenvalues are real. Moreover, it is a strictly positive-definite matrix, so that the system...

exists a positive definite symmetric matrix W such that W − GTWG is positive definite. Semipositivity and diagonal dominance A is semi-positive. That is...

the symmetric positive-definite matrix Σ = [ A B B T C ] , {\displaystyle \Sigma =\left[{\begin{matrix}A&B\\B^{\mathrm {T} }&C\end{matrix}}\right],} where...

real non-singular matrix can be uniquely factored as the product of an orthogonal matrix and a symmetric positive definite matrix, which is called a...

apparent. If A and B are both symmetric or Hermitian, and B is also a positive-definite matrix, the eigenvalues λi are real and eigenvectors v1 and v2 with distinct...

neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive. The table...

{\displaystyle U} is a p × p {\displaystyle p\times p} positive definite matrix with a matrix variate beta distribution, and a , b > ( p − 1 ) / 2 {\displaystyle...

neither positive-semidefinite nor negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is...

distribution with a nonsingular covariance matrix Σ precisely if Σ ∈ Rp × p is a positive-definite matrix and the probability density function of X is...

^{T}} matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the positive definite matrix M T...

mathematics, positive semidefinite may refer to: Positive semidefinite function Positive semidefinite matrix Positive semidefinite quadratic form Positive semidefinite...

Note that the Jacobi method does not converge for every symmetric positive-definite matrix. For example, A = ( 29 2 1 2 6 1 1 1 1 5 ) ⇒ D − 1 ( L + U ) =...

of probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions are of...

analysis, an incomplete Cholesky factorization of a symmetric positive definite matrix is a sparse approximation of the Cholesky factorization. An incomplete...

^{2}&\lambda \\\lambda &1\end{pmatrix}},} which is a positive definite matrix. A typical shear matrix is of the form S = ( 1 0 0 λ 0 0 1 0 0 0 0 0 1 0 0...

mathematics, particularly matrix theory, a Stieltjes matrix, named after Thomas Joannes Stieltjes, is a real symmetric positive definite matrix with nonpositive...

the system matrix A {\displaystyle A} is symmetric positive-definite, the splitting matrix M {\displaystyle M} is symmetric positive-definite, the stationary...