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Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and ...

A regular ring in the sense of commutative algebra is a commutative unit ring such that all its localizations at prime ideals are regular local rings. In contrast, a von ...

On a Lie group, exp is a map from the Lie algebra to its Lie group. If you think of the Lie algebra as the tangent space to the identity of the Lie group, exp(v) is defined ...

The Gelfand transform x|->x^^ is defined as follows. If phi:B->C is linear and multiplicative in the senses phi(ax+by)=aphi(x)+bphi(y) and phi(xy)=phi(x)phi(y), where B is a ...

Informally, the sample space for a given set of events is the set of all possible values the events may assume. Formally, the set of possible events for a given random ...

Let H be a Hilbert space and (e_i)_(i in I) is an orthonormal basis for H. The set S(H) of all operators T for which sum_(i in I)||Te_i||^2<infty is a self-adjoint ideal of ...

The quaternions are members of a noncommutative division algebra first invented by William Rowan Hamilton. The idea for quaternions occurred to him while he was walking along ...

A polynomial with matrix coefficients. An nth order matrix polynomial in a variable t is given by P(t)=A_0+A_1t+A_2t^2+...+A_nt^n, where A_k are p×p square matrices.

A periodic matrix with period 1, so that A^2=A.

The power A^n of a matrix A for n a nonnegative integer is defined as the matrix product of n copies of A, A^n=A...A_()_(n). A matrix to the zeroth power is defined to be the ...