Shear matrix

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. Such a matrix may be derived by taking the identity matrix and replacing one of the zero elements with a non-zero value.

A typical shear matrix is shown below:

The name shear reflects the fact that the matrix represents a shear transformation. Geometrically, such a transformation takes pairs of points in a linear space, that are purely axially separated along the axis whose row in the matrix contains the shear element, and effectively replaces those pairs by pairs whose separation is no longer purely axial but has two vector components. Thus, the shear axis is always an eigenvector of S.

A shear parallel to the x axis results in ${\displaystyle x'=x+\lambda y}$ and ${\displaystyle y'=y}$. In matrix form:

Similarly, a shear parallel to the y axis has ${\displaystyle x'=x}$ and ${\displaystyle y'=y+\lambda x}$. In matrix form:

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