ReflectionMatrix

Creates a reflection matrix.

Syntax

• ReflectionMatrix(v)

  • v is a vector

Description

If v is a non-zero vector, then ReflectionMatrix(v) returns the matrix for reflection in the hyperplane having v as a normal.

If w ≔ normalized(v) then

ReflectionMatrix(v) = IdentityMatrix(dim(v)) − 2⋅w⊗w

Examples

A ≔ ReflectionMatrix(❨1, 3, −2❩)

⎛ 0.857142857143  −0.428571428571   0.285714285714⎞
⎜−0.428571428571  −0.285714285714   0.857142857143⎟
⎝ 0.285714285714   0.857142857143   0.428571428571⎠

IsSymmetric(A) ∧ IsOrthogonal(A) ∧ IsInvolution(A) ∧ spectrum(A) = {−1, 1} ∧ det(A) = −1

true

See also

• OuterProduct (⊗)

• RotationMatrix