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...
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all eigenvalues are real. A symmetric n×n-matrix is called positive-definite (respectively negative-definite; indefinite), if for all nonzero vectors x...
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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...
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Positive-definite kernel Positive-definite matrix Positive-definite quadratic form Fasshauer, Gregory E. (2011), "Positive definite kernels: Past, present and...
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is unique and is called the inverse matrix of A, denoted A−1. Asymmetric real matrix A is called positive-definite if the associated quadratic form f (x)...
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non-matrix form will also contain some terms involving products of two different variables. Positive or negative-definiteness or semi-definiteness, or...
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Cholesky decomposition (category Matrix decompositions)
is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful...
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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...
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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;...
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Polar decomposition (redirect from Matrix polar decomposition)
semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix and P {\displaystyle P} is a positive semi-definite symmetric matrix in the...
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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...
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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...
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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...
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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...
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in particular: Negative-definite bilinear form Negative-definite quadratic form Negative-definite matrix Negative-definite function This article includes...
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definite pencil or definite pencil. Eigenvalue perturbation Frobenius covariant Householder transformation Jordan normal form List of matrices Matrix...
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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...
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algebra, a matrix decomposition or matrix factorization is a factorization of a matrix into a product of matrices. There are many different matrix decompositions;...
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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...
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^{T}} matrix inversion can be reduced to inverting symmetric matrices and 2 additional matrix multiplications, because the positive definite matrix M T...
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of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by...
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Wishart distribution (redirect from Wishart matrix)
probability distributions defined over symmetric, positive-definite random matrices (i.e. matrix-valued random variables). These distributions are of great...
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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 −...
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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...
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Shear mapping (redirect from Shear matrix)
^{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...
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Schur product theorem (category Matrix theory)
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...
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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...
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Determinant (redirect from Matrix determinant)
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...
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Trace (linear algebra) (redirect from Trace of a matrix)
B are real positive semi-definite matrices of the same size. The Frobenius inner product and norm arise frequently in matrix calculus and statistics....
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List of named matrices (redirect from List of matrix)
matrices used in mathematics, science and engineering. A matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers called entries...
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