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 -coordinate):
Vertical shear:
Example
Horizontal shear with applied to the unit square:
Square becomes a parallelogram. Area — preserved.
77.2. Properties
- Determinant: — area / volume preserving
- Inverse: shear by in the same direction:
- Eigenvalues: (repeated) — both with the same eigenvector for the horizontal shear; not diagonalizable when
- Fixed line: the axis being sheared along (e.g., the -axis stays put for horizontal shear)
- Triangular: shear matrices are upper / lower triangular
77.3. Why
Geometrically: a parallelogram slides — its base is unchanged, and its height (perpendicular distance) is unchanged. Area base height stays the same. The same logic generalizes to dimensions.
77.4. Composition of shears
The product of two shears (in the same direction) is another shear:
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 adds a multiple of one coordinate to another. The matrix is the identity with a single off-diagonal non-zero entry: