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...

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...

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...

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;...

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

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...

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...

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...

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...

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

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

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...

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 −...

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

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

{\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...

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...

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...

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...

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

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...