Foundations — QR decomposition

This page assumes you have seen nothing. We build every symbol the parent note uses, one at a time, each resting on the previous. Read top to bottom.

0. What is a vector, really?

The picture: a single straight arrow from the origin. Nothing more.

Why the topic needs it. The parent note writes ${\mathbf{a}}_1,{\mathbf{a}}_2,\dots$ — those are just the arrows we start with. Everything QR does is rearrange arrows, so "arrow = list of numbers" is line one.

1. A matrix as a stack of column-arrows

The picture: $n$ separate arrows, each described by $m$ numbers. The parent's example $A$ has $m=3$ (arrows in 3D space) and $n=2$ (two arrows).

Why the topic needs it. QR is a statement about columns. Reading $A$ as a stack of column-arrows is the whole mental switch that makes the rest visual.

2. Length of an arrow: the norm $\Vert \mathbf{a}\Vert$

Why the topic needs it. The parent writes $r_{kk}=\Vert {\mathbf{u}}_k\Vert$ and "normalize" — both are about length. You cannot shrink an arrow to length 1 without first knowing its length.

3. The dot product ${\mathbf{q}}^{\top }\mathbf{a}$ — the single most important tool here

Why the topic needs it. Every $r_{ik}={\mathbf{q}}_i^{\top }{\mathbf{a}}_k$ in the parent note is a dot product. And "orthonormal columns," "$Q^{\top }Q=I$," and "perpendicular" all cash out as dot products being zero. Learn this and half of QR is already yours.

4. Projection — the "shadow" of one arrow on another

Let us actually check that last claim, because it is the trick the whole parent page runs on. Take the dot product of $\mathbf{q}$ with $\mathbf{u}$: ${\mathbf{q}}^{\top }\mathbf{u}={\mathbf{q}}^{\top }\mathbf{a}-({\mathbf{q}}^{\top }\mathbf{a})\underset{=1}{\underbrace{{\mathbf{q}}^{\top }\mathbf{q}}}={\mathbf{q}}^{\top }\mathbf{a}-{\mathbf{q}}^{\top }\mathbf{a}=0.$ What we did: subtracted the shadow. Why: to kill any overlap with $\mathbf{q}$. What it looks like: the leftover arrow $\mathbf{u}$ stands at a perfect right angle to $\mathbf{q}$. This is exactly the parent's "remove what's already explained" step.

Why the topic needs it. The parent's ${\mathbf{u}}_2={\mathbf{a}}_2-({\mathbf{q}}_1^{\top }{\mathbf{a}}_2){\mathbf{q}}_1$ is subtract-the-shadow. Projection is the engine of Gram–Schmidt, which is the engine of QR — see Orthonormal bases and projections.

5. Orthonormal — perpendicular and length 1

The picture: arrows like the clean corner of a room — up, right, forward — each of length 1, all at right angles. That is the "clean coordinate frame" the parent keeps mentioning.

Why the topic needs it. The $Q$ in $A=QR$ has orthonormal columns. That is what makes $Q$ preserve lengths and angles (a rotation/reflection).

6. The identity matrix $I$ and what $Q^{\top }Q=I$ means

Why the topic needs it. This is the parent's definition of $Q$. Now you can read it instead of memorizing it.

7. Upper-triangular — zeros below the diagonal

Why the topic needs it. $R$ in $A=QR$ is upper-triangular. The parent proves why (each ${\mathbf{a}}_k$ uses only earlier $\mathbf{q}$'s); here you just need to recognize the shape. The sibling LU decomposition uses triangular matrices too.

8. Linear independence — "no arrow is redundant"

The picture: three arrows that genuinely reach into three different directions, not all squashed onto one line or one plane.

Why the topic needs it. It is the one requirement in the parent's definition ($A$ has linearly independent columns). This is where Gram-Schmidt process would break down without it.

Prerequisite map

Read it as: arrows feed everything; the dot product feeds projection and orthonormality; those two build $Q$; leftover lengths build $R$; together they are $A=QR$.

Equipment checklist

Test yourself — each line is question ::: answer. Cover the right side.

When every line above is easy, head back to QR decomposition and the derivation will read like plain English.