Shear Matrix

77. Shear Matrix

A shear matrix shifts each point in one direction by an amount proportional to its coordinate in another. The result: parallelograms slide into other parallelograms, preserving area.

77.1. 2D shears

Horizontal shear (each point shifts horizontally by a multiple of its $𝑦$ -coordinate):

𝑆_{𝑥} (𝑘) = [\begin{matrix} 1 & 𝑘 \\ 0 & 1 \end{matrix}]

𝑆_{𝑥} (𝑘) [\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}] = [\begin{matrix} 𝑥 + 𝑘 𝑦 \\ 𝑦 \end{matrix}]

Vertical shear:

𝑆_{𝑦} (𝑘) = [\begin{matrix} 1 & 0 \\ 𝑘 & 1 \end{matrix}]

𝑆_{𝑦} (𝑘) [\begin{matrix} 𝑥 \\ 𝑦 \end{matrix}] = [\begin{matrix} 𝑥 \\ 𝑦 + 𝑘 𝑥 \end{matrix}]

Example

Horizontal shear with $𝑘 = 1$ applied to the unit square:

[\begin{matrix} 0 \\ 0 \end{matrix}] \to [\begin{matrix} 0 \\ 0 \end{matrix}] [\begin{matrix} 1 \\ 0 \end{matrix}] \to [\begin{matrix} 1 \\ 0 \end{matrix}] [\begin{matrix} 1 \\ 1 \end{matrix}] \to [\begin{matrix} 2 \\ 1 \end{matrix}] [\begin{matrix} 0 \\ 1 \end{matrix}] \to [\begin{matrix} 1 \\ 1 \end{matrix}]

Square becomes a parallelogram. Area $= 1$ — preserved.

77.2. Properties

Determinant: $det (𝑆) = 1$ — area / volume preserving
Inverse: shear by $- 𝑘$ in the same direction:

𝑆_{𝑥} {(𝑘)}^{- 1} = 𝑆_{𝑥} (- 𝑘) = [\begin{matrix} 1 & - 𝑘 \\ 0 & 1 \end{matrix}]

Eigenvalues: $1$ (repeated) — both with the same eigenvector $[\begin{matrix} 1 \\ 0 \end{matrix}]$ for the horizontal shear; not diagonalizable when $𝑘 \neq 0$
Fixed line: the axis being sheared along (e.g., the $𝑥$ -axis stays put for horizontal shear)
Triangular: shear matrices are upper / lower triangular

77.3. Why $det = 1$

Geometrically: a parallelogram slides — its base is unchanged, and its height (perpendicular distance) is unchanged. Area $=$ base $\times$ height stays the same. The same logic generalizes to $𝑛$ dimensions.

77.4. Composition of shears

The product of two shears (in the same direction) is another shear:

𝑆_{𝑥} (𝑘_{1}) 𝑆_{𝑥} (𝑘_{2}) = 𝑆_{𝑥} (𝑘_{1} + 𝑘_{2})

Two shears in different directions can produce non-shear results (composition need not be triangular).

77.5. Application: graphics and italics

Italic typography: italic letters are upright letters with a horizontal shear applied
Graphics transforms: any 2D affine transformation can be decomposed into translate, rotate, scale, shear
Solving linear systems: row operations are shears applied to the matrix — that’s why row reduction preserves the row-space dimensions but reshapes the picture

77.6. Shears in higher dimensions

A general shear in $ℝ^{𝑛}$ adds a multiple of one coordinate to another. The matrix is the identity with a single off-diagonal non-zero entry:

𝐼 + 𝑘 𝑒_{𝑖} 𝑒_{𝑗}^{𝑇}, 𝑖 \neq 𝑗