Math 2.0

Linear and multilinear algebra; matrix theory Definition

Transvection

Builds on Elementary matrix

Definition

Let RR be a commutative ring, let n≥2n \ge 2, let i≠ji \ne j be indices in {1,…,n}\{1, \dots, n\} and let c∈Rc \in R. Let EijE_{ij} be the n×nn \times n matrix with entry 11 in position (i,j)(i, j) and 00 elsewhere. The transvection matrix is Tij(c)=In+cEij.T_{ij}(c) = I_n + c E_{ij}.

Remarks

The product Tij(c)AT_{ij}(c) A adds cc times row jj of AA to row ii; this is an elementary matrix of the row-addition type. Its determinant is 11, and its inverse is Tij(−c)T_{ij}(-c). Over a field, the transvections generate the special linear group. In the plane, (1c01)\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix} is a horizontal shear.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Shear_mapping, lead section
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