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.

Applications

Shear matrices are often used in computer graphics.^[1]

Notes

↑ Foley et al. (1991, pp. 207–208,216–217)

References

Foley, James D.; van Dam, Andries; Feiner, Steven K.; Hughes, John F. (1991), Computer Graphics: Principles and Practice (2nd ed.), Reading: Addison-Wesley, ISBN 0-201-12110-7

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[1] Foley et al. (1991, pp. 207–208,216–217)

Computer graphics
Vector graphics	Diffusion curve Pixel
2D graphics	2.5D
3D graphics	Aliasing Graphical projection
Concepts	Affine transformation Planar projection Rotation Scaling Shear matrix Translation
Computer graphics

Matrix classes
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Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
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Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Orthonormal Singular Unimodular Unipotent Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Coxeter Derogatory Distance Duplication Elimination Euclidean distance Fundamental (linear differential equation) Generator Gramian Hessian Householder Jacobian Moment Payoff Pick Random Rotation Seifert Shear Similarity Symplectic Totally positive Transformation Wedderburn X–Y–Z
Used in statistics	Bernoulli Centering Correlation Covariance Design Dispersion Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Skew-adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan canonical form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Quaternionic matrix Row echelon form Wronskian
List of matrices Category:Matrices

Shear matrix

Properties

Applications

See also

Notes

References