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In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator acting on an inner product space is called positive-semidefinite (or non-negative) if, for every (), , and , , where is the domain of .
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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.
A Hermitian diagonally dominant matrix with real non-negative diagonal entries is positive semidefinite. This follows from the eigenvalues being real, and Gershgorin's circle theorem. If the symmetry requirement is eliminated, such a matrix is not necessarily positive semidefinite. For example, consider
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 English, negation is achieved by adding not after the verb. As a practical matter, Modern English typically uses a copula verb (a form of be) or an auxiliary verb with not. If no other auxiliary verb is present, then dummy auxiliary do (does, did) is normally introduced – see do-support. For example, (8) a. I have gone (affirmative) b.
If is positive definite, then the eigenvalues are all positive reals, so the chosen diagonal of also consists of positive reals. Hence the eigenvalues of Q U 1 2 Q ∗ {\displaystyle QU^{\frac {1}{2}}Q^{*}} are positive reals, which means the resulting matrix is the principal root of A {\displaystyle A} .
In operator theory, a branch of mathematics, a positive-definite kernel is a generalization of a positive-definite function or a positive-definite matrix. It was first introduced by James Mercer in the early 20th century, in the context of solving integral operator equations .