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Reflection

Fixed subspace, negated orthogonal complement

What does it mean to reflect across a subspace?

Reflection across a subspace S decomposes every vector into a component inside S and a component in S⊥. The component in S is preserved, while the component in S⊥ is negated. This gives the reflection matrix R = 2P − I, where P is the orthogonal projection matrix onto S.

Open chapter PDF Key ideas Chapter index Back to Linear Algebra

Key ideas

Reflection is easiest to understand through orthogonal decomposition.

  • Every vector v ∈ ℝⁿ can be written as v∥ + v⊥, with v∥ ∈ S and v⊥ ∈ S⊥
  • Reflection across S preserves v∥ and negates v⊥
  • If P is the orthogonal projection matrix onto S, then the reflection matrix is R = 2P − I
  • Applying the same reflection twice gives the original vector back, so R² = I and R is its own inverse
  • Reflection across a line in ℝ³ preserves one direction and negates two directions, so det(R) = 1
  • Reflection across a plane in ℝ³ preserves two directions and negates one direction, so det(R) = −1

The final section compares two common conventions: projecting onto the fixed subspace S, or projecting onto its orthogonal complement S⊥.

Why is the formula R = 2P − I?

Since Pv = v∥, the perpendicular component is v − Pv. Reflection keeps v∥ and subtracts v⊥, so the reflected vector is Pv − (v − Pv) = (2P − I)v.

Related chapters

  • Projection provides the orthogonal projection matrix P used in the reflection formula R = 2P − I
  • Linear transformations in 3D gives the geometric language for line and plane reflections acting on vectors and cubes
  • Determinant explains why preserving or reversing orientation appears as determinant 1 or −1
  • Inverse connects to the identity R² = I, which means every reflection matrix is its own inverse
  • Householder reflection applies reflection matrices to QR factorization and uses the complementary convention R = I − 2PS⊥

Chapter contents

The PDF is a single document. The page links below are best-effort: most browsers support them, but some viewers may ignore the page hint.

Topic Pages
Reflection across a subspace S ⊂ ℝⁿ 1–2
Reflection across a line spanned by u ⊂ ℝ³ 3–5
Reflection across a plane S ⊂ ℝ³ 5–7
The 2 conventions 7
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Why can a reflection have determinant 1 or −1?

The determinant sign depends on how many independent directions are negated. In ℝ³, reflection across a line preserves one direction and negates two directions, so orientation is preserved and det(R) = 1. Reflection across a plane preserves two directions and negates one direction, so orientation is reversed and det(R) = −1.

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