2×2 Orthogonal Matrix Mastery — A Generalised Construction

2×2 Orthogonal Matrix Mastery — A Generalised Construction

Orthogonal matrices in two dimensions reveal one of the cleanest structures in linear algebra. A 2×2 matrix is orthogonal when its columns (and rows) satisfy two conditions:

  • They are perpendicular (their dot product is zero);
  • They have unit length (their magnitude is one).

This article presents a clear generalisation: any pair of perpendicular vectors with equal magnitude can be normalised to form an orthogonal matrix.

2×2 Orthogonal Matrix Mastery — A Generalised Construction


1. Begin with two perpendicular vectors

Let the first vector be: 

(a, b)

A perpendicular vector can be chosen as: 

(−b, a)

Their dot products confirm orthogonality: 

(a, b) · (−b, a) = −ab + ab = 0
(a, −b) · (b, a) = ab − ab = 0

2. Compute their shared magnitude

Both vectors have the same length: 

|(a, b)| = √(a² + b²)

We can therefore normalise each one by dividing by √(a² + b²).

3. Form the matrix using the normalised vectors

Place the two normalised vectors as the columns of a matrix: 

(1 / √(a² + b²)) ·
[  a   −b ]
[  b    a ]

4. Verify orthogonality

Multiply the matrix by its transpose: 

(1 / (a² + b²)) ·
[  a   −b ] [  a    b ]
[  b    a ] [ −b    a ]

Carrying out the multiplication yields: 

(1 / (a² + b²)) ·
[ a² + b²      0     ]
[    0      a² + b²  ]

Dividing through by (a² + b²) gives the identity matrix: 

[ 1   0 ]
[ 0   1 ]

This confirms that the matrix is orthogonal.

Conclusion

Any pair of perpendicular vectors with the same magnitude can be normalised to produce a 2×2 orthogonal matrix. This general construction offers a clean and geometric viewpoint of rotations, reflections, and other transformations in two dimensions.

Adapted from handwritten notes for educational use.

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