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The class of copositive matrices can be characterized using principal submatrices. One such characterization is due to Wilfred Kaplan: [6]. A real symmetric matrix A is copositive if and only if every principal submatrix B of A has no eigenvector v > 0 with associated eigenvalue λ < 0.
Positive semidefinite matrix; Positive semidefinite quadratic form; Positive semidefinite bilinear form This page was last edited on 2 ... Cookie statement; Mobile view;
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:
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.
For example, a 2,1 represents the element at the second row and first column of the matrix. In mathematics, a matrix (pl.: matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.
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 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).}
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