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In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector , where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: Negative-definite bilinear form; Negative-definite quadratic form; Negative-definite matrix; Negative-definite function
For example, the matrix above is ... A is negative-definite; if f does produce both negative and positive ... some equation by piecewise linear functions, where the ...
In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every non-zero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite.
More generally, every complex skew-symmetric matrix can be written in the form = where is unitary and has the block-diagonal form given above with still real positive-definite. This is an example of the Youla decomposition of a complex square matrix.
For example, a matrix is often used to represent the coefficients in a system of ... the determinant is an n-linear function. ... For a positive definite matrix A, ...
For a function of n variables, the number of negative eigenvalues of the Hessian matrix at a critical point is called the index of the critical point. A non-degenerate critical point is a local maximum if and only if the index is n , or, equivalently, if the Hessian matrix is negative definite ; it is a local minimum if the index is zero, or ...
If the diagonal elements of D are real and non-negative then it is positive semidefinite, and if the square roots are taken with the (+) sign (i.e. all non-negative), the resulting matrix is the principal root of D. A diagonal matrix may have additional non-diagonal roots if some entries on the diagonal are equal, as exemplified by the identity ...