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In mathematics, positive semidefinite may refer to: Positive semidefinite function; ... This page was last edited on 2 May 2021, at 17:28 (UTC).
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an iterative algorithm , applicable to sparse systems that are too large to be handled by a direct ...
The Gram matrix is positive semidefinite, and every positive semidefinite matrix is the Gramian matrix for some set of vectors. The fact that the Gramian matrix is positive-semidefinite can be seen from the following simple derivation:
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In contrast with the complex case, a positive-semidefinite operator on a real Hilbert space may not be symmetric. As a counterexample, define A : R 2 → R 2 {\displaystyle A:\mathbb {R} ^{2}\to \mathbb {R} ^{2}} to be an operator of rotation by an acute angle φ ∈ ( − π / 2 , π / 2 ) . {\displaystyle \varphi \in (-\pi /2,\pi /2).}
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 ...
Positive-definiteness arises naturally in the theory of the Fourier transform; it can be seen directly that to be positive-definite it is sufficient for f to be the Fourier transform of a function g on the real line with g(y) ≥ 0.
The determinant of this matrix is −1, as the area of the green parallelogram at the right is 1, but the map reverses the orientation, since it turns the counterclockwise orientation of the vectors to a clockwise one. The determinant of a square matrix A (denoted det(A) or | A |) is a number encoding certain properties of the matrix.