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Deutsch: Dieses Dokument listet 20323 Symbole und die dazugehörigen LaTeX-Befehle auf. Manche Symbole sind in jedem LaTeX-2ε-System verfügbar; andere benötigen zusätzliche Schriftarten oder Pakete, die nicht notwendig in jeder Distribution mitgeliefert werden und daher selbst installiert werden müssen.
Print/export Download as PDF; Printable version; In other projects ... In mathematics, positive semidefinite may refer to: Positive semidefinite function ...
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:
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 .
According to that sign, the quadratic form is called positive-definite or negative-definite. A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "never negative" and "never positive", respectively.
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.
A positive semidefinite matrix S gives rise to an infinite-dimensional Kac–Moody algebra of affine type, or an affine Lie algebra. An indefinite matrix S gives rise to a Kac–Moody algebra of indefinite type. Since the diagonal entries of C and S are positive, S cannot be negative definite or negative semidefinite.
L is positive-semidefinite (that is for all ). This can be seen from the fact that the Laplacian is symmetric and diagonally dominant. L is an M-matrix (its off-diagonal entries are nonpositive, yet the real parts of its eigenvalues are nonnegative). Every row sum and column sum of L is zero. Indeed, in the sum, the degree of the vertex is ...