Section 39.4 Examples
Here we will provide two examples of constructing product-preserving matrices.Example 39.4.1. An orthogonal matrix.
The vectors
v1=[1100],v2=[−1102],v3=[−110−1],v4=[0010]
form an orthogonal basis of R4.
We can normalize these vectors into an orthonormal basis:
v′1=[1√21√200],v′2=[−1√61√602√6],v′3=[−1√31√30−1√3],v′4=[0010].
Placing these four vectors as columns in a 4×4 matrix results in an orthogonal matrix:
A=[1√2−1√6−1√301√21√61√30000102√6−1√30].
Example 39.4.2. A unitary matrix.
The vectors
v1=[11ii],v2=[1−3ii],v3=[i010],v4=[i0−12]
form an orthogonal basis of C4.
We can normalize these vectors into an orthonormal basis:
v′1=[1212i2i2],v′2=[12√3−√32i2√3i2√3],v′3=[i√201√20],v′4=[i√60−1√62√6].
Placing these four vectors as columns in a 4×4 matrix results in a unitary matrix:
A=[1212√3i√2i√612−√3200i2i2√31√2−1√6i2i2√302√6].