In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number z T M z {\displaystyle z^{\operatorname...

all eigenvalues are real. A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x...

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

is unique and is called the inverse matrix of A, denoted A−1. A symmetric real matrix A is called positive-definite if the associated quadratic form f (x)...

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

non-matrix form will also contain some terms involving products of two different variables. Positive or negative-definiteness or semi-definiteness, or...

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

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

algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;...

skew-symmetric matrix then A 2 {\textstyle A^{2}} is a symmetric negative semi-definite matrix. As a result of the first two properties above, the set of all skew-symmetric...

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 Hermitian;...

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

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...

Hermitian positive semi-definite matrix. Furthermore, if B {\displaystyle B} is row full-rank, then A {\displaystyle A} is positive definite. The entries on the...

definite pencil or definite pencil. Eigenvalue perturbation Frobenius covariant Householder transformation Jordan normal form List of matrices Matrix...

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...

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...

2\times 2} matrix would be necessary to fully characterize the two-dimensional variation. Any covariance matrix is symmetric and positive semi-definite and its...

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

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

in particular: Negative-definite bilinear form Negative-definite quadratic form Negative-definite matrix Negative-definite function This article includes...

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

is positive. There exists a symmetric positive definite matrix W such that AW + WAT is positive definite. A + I is non-singular, and G = (A + I)−1(A −...

states that the Hadamard product of two positive definite matrices is also a positive definite matrix. The result is named after Issai Schur (Schur 1911...

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 ) = ( 0...

j-th column of X {\displaystyle X} . The scatter matrix is the m-by-m positive semi-definite matrix S = ∑ j = 1 n ( x j − x ¯ ) ( x j − x ¯ ) T = ∑ j...

^{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 0 0...

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...

considered as a matrix over the real numbers, the following additional properties hold: J is positive semi-definite matrix. The matrix 1 n J {\displaystyle...

square matrix. The determinant of a matrix A is commonly denoted det(A), det A, or |A|. Its value characterizes some properties of the matrix and the...