In mathematics, an affine algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizable group defined over a field k is said to split overk or k-split if the isomorphism is defined over k. This coincides with the usual notion of split for an algebraic group. Every diagonalizable group splits over the separable closure ks of k. Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense.
The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian groups with Gal(ks/k)-equivariant morphisms without p-torsion, if k is of characteristic p. This is an analog of Poincaré duality and motivated the terminology.
A diagonalizable k-group is said to be anisotropic if it has no nontrivial k-valued character.
The so-called "rigidity" states that the identity component of the centralizer of a diagonalizable group coincides with the identity component of the normalizer of the group. The fact plays a crucial role in the structure theory of solvable groups.
A connected diagonalizable group is called an algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup.
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algebraic group is said to be diagonalizable if it is isomorphic to a subgroup of Dn, the group of diagonal matrices. A diagonalizablegroup defined over...
In linear algebra, a square matrix A {\displaystyle A} is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there...
abelian group A, one can form the corresponding diagonalizablegroup D(A), defined as a functor by setting D(A)(T) to be the set of abelian group homomorphisms...
matrix is diagonalizable. A matrix A = [ a b c d ] {\displaystyle A={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} belongs to the orthogonal group if AQAT =...
can be proved using the action of the twofold diagonal subgroup. Diagonalizablegroup Sahai, Vivek; Bist, Vikas (2003), Algebra, Alpha Science Int'l Ltd...
converse is also true; that is, if two diagonalizable matrices commute, they are simultaneously diagonalizable. But if you take any two matrices that...
algebraic groups requires more global tools. For an algebraically closed field k, a matrix g in GL(n,k) is called semisimple if it is diagonalizable, and unipotent...
3D manifold of rotation matrices. A method for finding log A for a diagonalizable matrix A is the following: Find the matrix V of eigenvectors of A (each...
means that in particular every ρ ( s ) {\displaystyle \rho (s)} is diagonalizable. For more details see the article on unitary representations. A representation...
{\mathfrak {sl}}(2,\mathbb {C} )} are traceless. But then Q is diagonalizable, hence q is diagonalizable, which is a contradiction. Rossmann 2002, Proposition...
for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic groups, a group element is unipotent...
matrix B. For example, A is called diagonalizable if it is similar to a diagonal matrix. Not all matrices are diagonalizable, but at least over the complex...
\operatorname {ad} (x):{\mathfrak {g}}\to {\mathfrak {g}}} is semisimple (i.e., diagonalizable). Sometimes this characterization is simply taken as the definition...
has n linearly independent eigenvectors. Such matrices are said to be diagonalizable. Over the field of real or complex numbers, more is true. The spectral...
linearly independent eigenvectors. Not all matrices are diagonalizable; matrices that are not diagonalizable are called defective matrices. Consider the following...
&e^{a_{n}}\end{bmatrix}}.} This result also allows one to exponentiate diagonalizable matrices. If A = UDU−1 and D is diagonal, then eA = UeDU−1. Application...
of matrices Shaw, Ronald (1982) Linear Algebra and Group Representations, v. 1, § 2.3, Diagonalizable linear operators, pages 78–81, Academic Press ISBN 0-12-639201-3...
are said to be diagonalizable. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after extending...
eigenvectors of A form a basis if and only if A is diagonalizable. A matrix that is not diagonalizable is said to be defective. For defective matrices,...
general linear Lie algebra all of whose elements are semisimple (or diagonalizable over an algebraically closed field). Equivalently, a Lie algebra is...
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