Foundations — Orthogonal matrices — properties, det = ±1

Before you can read a single line of the parent note, you need to earn every piece of notation. We build them one at a time, each on top of the last, from absolute zero. Nothing below assumes you have seen matrices before.

1. A vector — an arrow with an address

The picture: think of a treasure map instruction — "3 steps east, 4 steps north". That instruction is the vector $\left(\begin{array}{c}3\\ 4\end{array}\right)$. The arrow is the straight shortcut from where you started to the treasure.

Why the topic needs it: orthogonal matrices are machines that move vectors. To talk about "moving" something, we first need a name for the thing being moved.

2. Length — how long the arrow is, written $\Vert x\Vert$

Why the square root of squares? Look at the right triangle the arrow makes with the horizontal and vertical. Its two legs are $x_1$ and $x_2$; the arrow is the hypotenuse. Pythagoras — "the long side squared equals the sum of the leg squares" — hands us the formula directly. The square root undoes the squaring so we get an actual distance, not a distance-squared.

Why the topic needs it: the whole point of orthogonal matrices is that they preserve length ($\Vert Qx\Vert =\Vert x\Vert$). You cannot appreciate "length is preserved" until you can compute length.

3. The dot product — a number that measures agreement, written $x^{\mathsf{T}}y$

Why this tool and not another? We want ONE number that tells us whether two arrows point the same way, opposite ways, or perpendicular. The dot product is exactly that detector:

• if it is positive, the arrows broadly agree (angle less than $90^{\circ }$),
• if it is zero, they are exactly perpendicular (at a right angle),
• if it is negative, they broadly disagree (angle more than $90^{\circ }$).

The picture and the master formula: the dot product hides an angle inside it: $x^{\mathsf{T}}y=\Vert x\Vert \Vert y\Vert \cos \theta ,$ where $\theta$ is the angle between the arrows. When the arrows are perpendicular, $\theta =90^{\circ }$, $\cos 90^{\circ }=0$, so the whole product is $0$. That is why "dot product $=0$" is the same as "perpendicular".

Why the topic needs it: "columns are perpendicular" and "columns have length 1" — the two halves of orthonormal — are both statements about dot products.

4. The matrix $Q$ — a grid that stores where the arrows go

We write $Qx$ for "apply the machine $Q$ to the arrow $x$". The parent's notation $q_1,q_2,\dots ,q_n$ means "the columns of $Q$", written $Q=[q_1\mid q_2\mid \cdots \mid q_n]$.

Why the topic needs it: the entire subject is about a special family of these grids. The columns are where all the geometry lives.

5. The transpose $Q^{\mathsf{T}}$ — flip across the diagonal

Why the little ${}^{\mathsf{T}}$ appears in $x^{\mathsf{T}}y$: a plain column vector is a tall $2\times 1$ matrix. Transposing it lays it flat into a $1\times 2$ row. Placing a flat row next to a tall column and multiplying "row-into-column" produces exactly one number — the dot product. So $x^{\mathsf{T}}y$ is literally "lay $x$ flat, then combine with $y$", giving $x_1{y}_1+x_2{y}_2$. The transpose is the notation that turns "dot product" into ordinary matrix multiplication.

Why the topic needs it: the definition of orthogonal is $Q^{\mathsf{T}}Q=I$. No transpose, no topic.

6. The identity $I$ and Kronecker delta ${\delta }_{ij}$ — the "do nothing" symbols

Why the topic needs it: the compact statement "$q_i^{\mathsf{T}}{q}_j={\delta }_{ij}$" says both orthonormal conditions in one line: same column dotted with itself is $1$ (unit length), different columns dotted give $0$ (perpendicular).

7. The determinant $\det Q$ — the signed area-scaling number

Why "signed"? The sign records orientation. If the two columns keep their original left-to-right ordering (like the original axes), the sign is $+$. If the machine has flipped the plane over — swapped the sense of turning — the sign is $-$. This is exactly the $+1=$ rotation, $-1=$ reflection story from the parent note.

Why the topic needs it: the headline result $\det Q=\pm 1$ is a statement about area. A rigid motion cannot change area, so the scaling factor can only be "keep area, same sign" ($+1$) or "keep area, flipped" ($-1$).

8. Eigenvalue $\lambda$ and $|\lambda |$ — the special stretch of a special arrow

$|\lambda |$ means the size of $\lambda$ ignoring sign (or, for complex numbers, its distance from zero). For an orthogonal matrix, no length ever changes, so $|\lambda |=1$ always — eigenvalues sit on the unit circle. You will meet these again in Eigenvalues and eigenvectors.

Why the topic needs it: one of the parent's key properties is "all eigenvalues have $|\lambda |=1$" — a direct echo of length-preservation.

How the foundations feed the topic

Each foundation you built points forward: lengths and dot products define orthonormal; matrix and transpose write the defining condition; the determinant delivers the $\pm 1$ headline; eigenvalues give the unit-circle property. Related deeper roads branch off to Orthonormal bases & Gram–Schmidt, Rotations and reflections in $\mathbb{R}^2$ and $\mathbb{R}^3$, and Determinants — properties.

Equipment checklist

Test yourself — cover the right side and answer before revealing.