Search results
Results from the WOW.Com Content Network
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:
The Matrix Market exchange formats are a set of human readable, ASCII-based file formats designed to facilitate the exchange of matrix data. The file formats were designed and adopted for the Matrix Market, a NIST repository for test data for use in comparative studies of algorithms for numerical linear algebra. [1]
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 ...
Upload file; Special pages; Permanent link; Page information; Cite this page; Get shortened URL; Download QR code; ... In mathematics, positive semidefinite may refer to:
Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Donate
This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point).
We say that A ≥ B if A − B is positive semi-definite. Similarly, we say that A > B if A − B is positive definite. Although it is commonly discussed on matrices (as a finite-dimensional case), the Loewner order is also well-defined on operators (an infinite-dimensional case) in the analogous way.
As the transposition map preserves eigenvalues, the spectrum of () is the same as the spectrum of , and in particular () must still be positive semidefinite. Thus must also be positive semidefinite. This proves the necessity of the PPT criterion.