Gram-Schmidt builds orthogonal directions one by one, then normalizes them to create Q. This chapter develops that process, shows why Q preserves lengths and angles, explains the determinant and geometric meaning of square Q, extends the picture to tall Q and derives QQᵀ as the orthogonal projection onto col(Q).
Gram-Schmidt removes components along earlier directions so that each new vector contributes only its genuinely new direction. Normalizing those directions produces the columns of Q.
Part 2 then turns from Q to R and shows how QR factorization separates deformation from rotation or reflection.
Because Q preserves lengths and angles. In the square case it acts as a rigid motion, and in the tall case it embeds the domain into a higher-dimensional space without distortion.
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| Topic | Pages |
|---|---|
| Gram-Schmidt orthogonalization | 1–2 |
| Structure of matrix Q | 3–4 |
| Properties of Q defined as QᵀQ = I | 4–5 |
| Determinant of square Q | 5–6 |
| Combined rotation and reflection | 6–8 |
| Transformation by a 'tall' Q (m>n) | 8–10 |
| Matrix Q Qᵀ | 10–11 |
| Q Qᵀ as sum of outer products | 11–14 |
Because when the columns of Q are orthonormal, QᵀQ = I and the general projection formula simplifies to Q(QᵀQ)⁻¹Qᵀ = QQᵀ. This makes QQᵀ the orthogonal projection onto the space spanned by the columns of Q.