elementary matrix

The matrix A can be expressed as a finite product of elementary matrices.

The elementary matrices generate the general linear group of invertible matrices.

The elementary row operations may be viewed as the multiplication on the left of the original matrix by elementary matrices.

Similarly, K(R) is a modification of the group of units in a ring, using elementary matrix theory.

In algebraic K-theory, "elementary matrices" refers only to the row-addition matrices.

As the elementary matrices generate the commutator subgroup, this map is onto the commutator subgroup.

In mathematics, an elementary matrix is a matrix which differs from the identity matrix by one single elementary row operation.

Transforming into its reduced row echelon form amounts to left-multiplying by a matrix which is a product of elementary matrices, so , where .

In mathematics, a shear matrix or transvection is an elementary matrix that represents the addition of a multiple of one row or column to another.

The corresponding elementary matrix is a diagonal matrix, with diagonal entries 1 everywhere except in the ith position, where it is m.

There are three types of elementary matrices, which correspond to three types of row operations (respectively, column operations):

One of the three classes of elementary matrix is involutory, namely the row-interchange elementary matrix.

An elementary matrix here is a transvection: one such that all main diagonal elements are 1 and there is at most one non-zero element not on the diagonal.

The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group is the group generated by elementary matrices.

Left multiplication (pre-multiplication) by an elementary matrix represents elementary row operations, while right multiplication (post-multiplication) represents elementary column operations.

The subgroup generated by elementary matrices is exactly the derived subgroup, in other words the smallest normal subgroup such that the quotient by it is abelian.

Both methods proceed by multiplying the matrix by suitable elementary matrices, which correspond to permuting rows or columns and adding multiples of one row to another row.

Some mathematics books use U and E to represent the Identity Matrix (meaning "Unit Matrix" and "Elementary Matrix", or from the German "Einheitsmatrix", respectively), although I is considered more universal.

The argument for is similar to the case, only one replaces Tietze moves for the fundamental group presentation by elementary matrix operations for the presentation matrices for (using the presentation matrices coming from cellular homology.

In other words, the Whitehead group Wh(G) of a group G is the quotient of GL(Z[G]) by the subgroup generated by elementary matrices, elements of G and 1.

The final type of row operation on a matrix A adds row j multiplied by a scalar m to row i. The corresponding elementary matrix is the identity matrix but with an m in the (i,j) position.

A special case of another class of elementary matrix, that which represents multiplication of a row or column by 1, is also involutory; it is in fact a trivial example of a signature matrix, all of which are involutory.

In this respect, the book is a sequel to the earlier work Elementary Matrices by Frazer, Duncan and Collar, a book which, by presenting problems in a form which could be assimilated by computers, stimulated the growth of the latter.

Here Z[G] is the group ring of G. Recall that the K-group K(A) of a ring A is defined as the quotient of GL(A) by the subgroup generated by elementary matrices.

At the height of this committee and other administrative work, Roderick often expressed his desire to return to his mathematics and, in particular, to write a sequel to his famous book (with R.A. Frazer and W.J. Duncan) Elementary Matrices.