An example of a 4x4 Haar transformation matrix is shown below.

For this reason, 4x4 transformation matrices are widely used in 3D computer graphics.

In two dimensions, linear transformations can be represented using a 2x2 transformation matrix.

The basis transformation matrix can be regarded as an automorphism over V.

In this case, transformations are obtained with a post-multiplication by a transformation matrix.

Therefore, any linear transformation can be also represented by a general transformation matrix.

The transformation matrix should account for power in variance in the two frames of reference.

If the L2 norm of and is unity, the transformation matrix can be expressed as:

The transformation matrix is proper orthogonal in order to allow rotations but no reflections.

The components of a tensor transform in a similar manner with a transformation matrix for each index.

Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is -1.

The transformation matrix is universal for all four-vectors, not just 4-dimensional spacetime coordinates.

Thus two types of transformation matrices are consistent with group postulates:

Translation and rotation combined in one transformation matrix is:

Transformation matrix is a matrix containing the coefficients that relates new and old variables.

We can write as two separate transformation matrices:

Such effects are most easily characterized in the form of a complex 2x2 transformation matrix called the Jones matrix:

Assuming that translation is not used transformations can be combined by simply multiplying the associated transformation matrices.

Replacing v with v in the transformation matrix gives:

Further, rotations preserve the quantum mechanical inner product, and so should our transformation matrices:

Such a matrix is also called 'atomic' upper/lower triangular or 'Gauss transformation matrix'.

Then use the transformation matrix:

Here each symmetry operation will either leave the motion unchanged or reverse its direction, so the transformation matrices are simply + 1 or -1, respectively.

Two things to note regarding the RGB transformation matrix:

There is an alternative expression of transformation matrices involving row vectors that is preferred by some authors.