Matrices, algebraic

Calculation of the algebraic matrices (the angular parts of the energy matrices)... 

Differential Calculus Integral Calculus Special Functions Basic Definitions Linear Algebra Matrices... 

An introduction to matrix algebra matrices, vectors and determinants... 

Moore EH (1920) On the reciprocal of the general algebraic matrix. Bull Amer Math Soc 26 394... 

The above set of differential equations (20 equations) can be solved with the help of linear algebra (matrix operation). Even though the math is not particularly ... 

First we try to simplify the two equations by algebraic matrix manipulations. 

Research Areas Linear Algebra, Matrix Theory, Numerical Analysis, Numerical Algebra, Geometry, Krein Spaces, Graph Theory, Mechanics, Inverse Problems, Mathematical Education, Applied Mathematics, Geometric Computing. 

The nonequilibrium zeroth Green s functions are determined by the Dyson equations (62) and (63) on the Keldysh contour. The standard way to solve these equations is to perform a Fourier transform and then solve the algebraic matrix equations for the Green s functions. For the Keldysh functions, this procedure cannot be implemented in a straightforward way because of two time branches. Thus, we should find the Fourier transform for each Keldysh function after applying the Langreth s mapping procedure described in Section 2 [41, 45]. In particular for — t ), the Dyson... 

The basic fact we need is that on an algebraic matrix group S SL + t(/c) the functions xi bx, xh- x 1, and xi- x ibx for fixed b are continuous. This is clear, since they are given by polynomials, and polynomial maps are always continuous in the Zariski topology. It is worth mentioning only because multiplication is not jointly continuous (it is a continuous map S x S- S, but the topology on S x S is not the product topology). 

Condiary. Every affine algebraic group is isomorphic to an algebraic matrix group. 

Theorem. Let S be an algebraic matrix group. Let S° be the connected component containing the unit e. Then S° is a normal subgroup of finite index it is irreducible, and the other irreducible components are its cosets. 

We call G° the connected component of G. Unlike algebraic matrix groups, the G here need not have the other f( A isomorphic to A0 this fails in our introductory example of p3 over the reals. 

Theorem. Let g in an algebraic matrix group G(k) GL (fc) be separable. Then in any representation of G the element g acts as a separable transformation. 

Theorem. Let g be a unipotent element of an algebraic matrix group. Then g acts as a unipotent transformation in every linear representation. Homomar-phisms take unipotent elements to unipotent elements, and unipotence is an intrinsic property. 

If G comes from an algebraic matrix group, these are equivalent to ... 

Proof. Base-extending to k, we may assume we have a finite constant group scheme, say of order n. When we embed it as an algebraic matrix group, each g in it satisfies the separable equation X" — 1 = 0. If g is also unipotent, g — l. Thus the group is trivial. ... 

The argument at the start of the proof shows now that any homomorphism G-> H preserves Jordan decompositions. In particular, Jordan decomposition in an algebraic matrix group is intrinsic, independent of the choice of an embedding in GL . 

Theorem. Let k be perfect, S an abelian algebraic matrix group. Let Ss and Su be the sets of separable and unipotent elements in S. Then Ss and Su are closed subgroups, and S is their direct product. 

Let k be algebraically closed, G an algebraic matrix group. Show G is unipotent iff all elements of finite order have order divisible by char (k). [Use Kolchin s theorem to reduce to the abelian case, and look at diagonalizabie matrix groups.]... 

It is instructive to restate this proof geometrically for algebraic matrix groups. It first shows that the closure of the image of a product of connected sets is connected, then that the closure of the union of an increasing sequence of connected sets is connected. 

Corollary. Let S be a connected solvable algebraic matrix group. Then 2S is nilpotent. 

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