Editing Shear mapping (section)

==Definition==

===Horizontal and vertical shear of the plane===
{{refimprove section|date=April 2025}}
[[File:SVG skewX.svg|thumb|250px|Horizontal shear of a square into parallelograms with factors <math>\cot(60^\circ) = \tan(30^\circ) \approx 0.58</math> and <math>\cot(45^\circ) = \tan(45^\circ) = 1</math>]]
In the plane <math>\R^2 = \R\times\R</math>, a '''horizontal shear''' (or '''shear parallel''' to the {{mvar|x}}-axis) is a function that takes a generic point with coordinates <math>(x,y)</math> to the point <math>(x + m y,y)</math>; where {{mvar|m}} is a fixed parameter, called the '''shear factor'''.

The effect of this mapping is to displace every point horizontally by an amount proportionally to its {{mvar|y}}-coordinate.  Any point above the {{mvar|x}}-axis is displaced to the right (increasing {{mvar|x}}) if {{math|''m'' > 0}}, and to the left if {{math|''m'' < 0}}. Points below the {{mvar|x}}-axis move in the opposite direction, while points on the axis stay fixed.

Straight lines parallel to the {{mvar|x}}-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the {{mvar|x}}-axis. Vertical lines, in particular, become [[Angle#Types of angles|oblique]] lines with [[slope]] <math>\tfrac 1 m.</math> Therefore, the shear factor {{mvar|m}} is the [[cotangent]] of the '''shear angle''' <math>\varphi</math> between the former verticals and the {{mvar|x}}-axis.{{fact|date=April 2025}} In the example on the right the square is tilted by 30°, so the shear angle is 60°.

If the coordinates of a point are written as a [[column vector]] (a 2×1 [[matrix (mathematics)|matrix]]), the shear mapping can be written as [[matrix product|multiplication]] by a 2×2 matrix:
: <math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  =
  \begin{pmatrix}x + m y \\y \end{pmatrix} =
  \begin{pmatrix}1 & m\\0 & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

A '''vertical shear''' (or shear parallel to the {{mvar|y}}-axis) of lines is similar, except that the roles of {{mvar|x}} and {{mvar|y}} are swapped. It corresponds to multiplying the coordinate vector by the [[transpose of a matrix|transposed matrix]]:

:<math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  = 
  \begin{pmatrix}x \\ m x + y \end{pmatrix} = 
  \begin{pmatrix}1 & 0\\m & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

The vertical shear displaces points to the right of the {{mvar|y}}-axis up or down, depending on the sign of {{mvar|m}}.  It leaves vertical lines invariant, but tilts all other lines about the point where they meet the {{mvar|y}}-axis.  Horizontal lines, in particular, get tilted by the shear angle <math>\varphi</math> to become lines with slope {{mvar|m}}.

====Composition====
Two or more shear transformations can be combined.

If two shear matrices are <math display="inline">\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}</math> and <math display="inline">\begin{pmatrix} 1 & 0 \\ \mu & 1 \end{pmatrix}</math>

then their composition matrix is
<math display="block">\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \mu & 1\end{pmatrix} = \begin{pmatrix} 1 + \lambda\mu & \lambda \\ \mu & 1 \end{pmatrix},</math>
which also has determinant 1, so that area is preserved.

In particular, if <math>\lambda=\mu</math>, we have

<math display="block">\begin{pmatrix} 1 + \lambda^2 & \lambda \\ \lambda & 1 \end{pmatrix},</math>

which is a [[positive definite matrix]].

===Higher dimensions===
A typical shear matrix is of the form
<math display="block">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}.</math>

This matrix shears parallel to the {{mvar|x}} axis in the direction of the fourth dimension of the underlying vector space.

A shear parallel to the {{mvar|x}} axis results in <math>x' = x + \lambda y</math> and <math>y' = y</math>. In matrix form:
<math display="block">\begin{pmatrix} x' \\ y' \end{pmatrix} = 
\begin{pmatrix}
1 & \lambda \\
0 & 1
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}.</math>

Similarly, a shear parallel to the {{mvar|y}} axis has <math>x' = x</math> and <math>y' = y + \lambda x</math>. In matrix form:
<math display="block">\begin{pmatrix}x' \\ y' \end{pmatrix} =
\begin{pmatrix}
1 & 0 \\
\lambda & 1
\end{pmatrix}
\begin{pmatrix} x \\ y \end{pmatrix}.</math>

In 3D space this matrix shear the YZ plane into the diagonal plane passing through these 3 points: <math>(0, 0, 0)</math> <math>(\lambda, 1, 0)</math> <math>(\mu, 0, 1)</math> 
<math display="block">S = \begin{pmatrix}
1 & \lambda & \mu \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}.</math>

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 will not contribute to the determinant. Thus every shear matrix has an [[inverse matrix|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 {{mvar|S}} is a shear matrix with shear element {{math|&lambda;}}, then {{mvar|S<sup>n</sup>}} is a shear matrix whose shear element is simply {{math|''n''&lambda;}}. Hence, raising a shear matrix to a power {{mvar|n}} multiplies its [[shear mapping#Definition|shear factor]] by {{mvar|n}}.

====Properties====
If {{mvar|S}} is an {{math|''n'' × ''n''}} shear matrix, then:
* {{mvar|S}} has [[rank of a matrix|rank]] {{mvar|n}} and therefore is [[invertible matrix|invertible]]
* 1 is the only [[eigenvalue]] of {{mvar|S}}, so {{math|1=[[determinant|det]] ''S'' = 1}} and {{math|1=[[Trace (linear algebra)|tr]] ''S'' = ''n''}}
* the [[eigenspace]] of {{mvar|S}} (associated with the eigenvalue 1) has {{math|''n'' − 1}} dimensions.
* {{mvar|S}} is [[Defective matrix|defective]]
* {{mvar|S}} is asymmetric
* {{mvar|S}} may be made into a [[block matrix]] by at most 1 column interchange and 1 row interchange operation
* the [[area (geometry)|area]], [[volume (geometry)|volume]], or any higher order interior capacity of a [[polytope]] is invariant under the shear transformation of the polytope's vertices.

===General shear mappings===
For a [[vector space]] {{mvar|V}} and [[Linear subspace|subspace]] {{mvar|W}}, a shear fixing {{mvar|W}} translates all vectors in a direction parallel to {{mvar|W}}.

To be more precise, if {{mvar|V}} is the [[direct sum of vector spaces|direct sum]] of {{mvar|W}} and {{mvar|W&prime;}}, and we write vectors as

:<math>v=w+w'</math>

correspondingly, the typical shear {{mvar|L}} fixing {{mvar|W}} is
:<math>L(v) = (Lw+Lw') = (w+Mw') + w',</math>
where {{mvar|M}} is a linear mapping from {{mvar|W&prime;}} into {{mvar|W}}. Therefore in [[block matrix]] terms {{mvar|L}} can be represented as

:<math>\begin{pmatrix} I & M \\ 0 & I \end{pmatrix}. </math>