Shear Matrix — Definition, Formula & Examples

A shear matrix is a square matrix that, when multiplied by a vector, shifts one component of that vector by an amount proportional to another component. It 'slants' shapes without changing their area.

A shear matrix is an elementary matrix obtained from the identity matrix by adding a nonzero off-diagonal entry $k$ in position $(i, j)$ where $i \neq j$ . It represents a linear map that fixes all basis vectors except that the image of $\mathbf{e}_j$ gains a component $k\mathbf{e}_i$ . The determinant of every shear matrix equals 1, so the transformation is volume-preserving.

Key Formula

S = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}

Where:

$k$ = Shear factor — determines how much the x-component is displaced per unit of y
$S$ = The 2D horizontal shear matrix (shears parallel to the x-axis)

How It Works

To build a 2D shear matrix, start with the

2 \times 2

identity matrix and place the shear factor

k

in the off-diagonal position corresponding to the direction you want to displace. A horizontal shear uses position

(1,2)

, shifting the

x

-component based on

y

. A vertical shear uses position

(2,1)

, shifting

y

based on

x

. When you multiply the shear matrix by a column vector, the targeted component changes while the other stays fixed. In higher dimensions, the same idea applies: place

k

at row

i

, column

j

to shear the

i

-th coordinate proportionally to the

j

-th.

Worked Example

Problem: Apply a horizontal shear with factor k = 2 to the point (3, 1).

Step 1: Write the shear matrix with k = 2 in position (1,2).

S = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}

Step 2: Multiply the shear matrix by the column vector.

S\begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} 1 \cdot 3 + 2 \cdot 1 \\ 0 \cdot 3 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 5 \\ 1 \end{pmatrix}

Answer: The sheared point is (5, 1). The x-component shifted by 2·1 = 2, while the y-component stayed at 1.

Why It Matters

Shear matrices appear in LU decomposition, where Gaussian elimination steps are encoded as shear (elementary) matrices. In computer graphics, shearing is a basic affine operation used to skew images and derive oblique projections. Understanding shear also clarifies why row operations preserve determinant sign and magnitude.

Common Mistakes

Mistake: Placing the shear factor on the diagonal instead of off-diagonal.

Correction: The diagonal entries of a shear matrix are all 1. The shear factor k goes in an off-diagonal position (i, j) with i ≠ j. Changing the diagonal would introduce scaling, not shearing.