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Download QR code; Print/export ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ... Cookie statement; Mobile view;
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
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).}
If c 1 < 0 and c 2 < 0 , the quadratic form is negative-definite and always evaluates to a negative number whenever [,] [,] . And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number.
In coding theory, Hamming(7,4) is a linear error-correcting code that encodes four bits of data into seven bits by adding three parity bits. It is a member of a larger family of Hamming codes , but the term Hamming code often refers to this specific code that Richard W. Hamming introduced in 1950.
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.
There are two main relaxations of QCQP: using semidefinite programming (SDP), and using the reformulation-linearization technique (RLT). For some classes of QCQP problems (precisely, QCQPs with zero diagonal elements in the data matrices), second-order cone programming (SOCP) and linear programming (LP) relaxations providing the same objective value as the SDP relaxation are available.
For positive semidefinite A, an decomposition exists where the number of non-zero elements on the diagonal D is exactly the rank of A. [11] Some indefinite matrices for which no Cholesky decomposition exists have an LDL decomposition with negative entries in D : it suffices that the first n − 1 leading principal minors of A are non-singular.