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:

$S= \begin{pmatrix} 1 & 0 & 0 & \lambda & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{pmatrix}.$

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 x' = x + $\lambda$y and y' = y. In matrix form:

$\begin{pmatrix} x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}.$

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

$\begin{pmatrix}x' \\ y' \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ \lambda & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}.$

Clearly the determinant will always be 1, as no matter where the shear element is placed, it will be a member of a skew-diagonal that also contains zero elements (as all skew-diagonals have length at least two) hence its product will remain zero and won't contribute to the determinant. Thus every shear matrix has an inverse, and the inverse is simply a shear matrix with the shear element negated, representing a shear transformation in the opposite direction. In fact, this is part of an easily derived more general result: if S is a shear matrix with shear element $\lambda$, then Sn is a shear matrix whose shear element is simply n$\lambda$. Hence, raising a shear matrix to a power n multiplies its shear factor by n.

Properties

If S is an n×n shear matrix, then:

• S has rank n and therefore is invertible
• 1 is the only eigenvalue of S, so det S = 1 and trace S = n
• the eigenspace of S has n-1 dimensions.
• S is asymmetric
• S may be made into a block matrix by at most 1 column interchange and 1 row interchange operation
• the area, volume, or any higher order interior capacity of a polytope is invariant under the shear transformation of the polytope's vertices.