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

negative-semidefinite. A symmetric matrix is positive-definite if and only if all its eigenvalues are positive, that is, the matrix is positive-semidefinite and...

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

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

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

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

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...

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...

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;...

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...

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...

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...

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

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

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...

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

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 A...

invertible matrix (non-singular, non-degenarate or regular) is a square matrix that has an inverse. In other words, if some other matrix is multiplied...

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

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

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

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 −...

{\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...

such norms are referred to as matrix norms. Matrix norms differ from vector norms in that they must also interact with matrix multiplication. Given a field...

The p×p matrix Qxx = E[ xi xiT ] is of full rank, and hence positive-definite; {xiεi} is a martingale difference sequence, with a finite matrix of second...

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...

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...