# Physical transformation associated with a Pseudo-Orthogonal matrix [closed]

An orthogonal matrix $$O$$, which belongs to an orthogonal group, is characterized as $$O^TO=I$$. Let's take an example of a $$2 \times 2$$ orthogonal matrix, $$O = \begin{bmatrix} \phantom{-} \cos{\theta} & \sin{\theta} \\ - \sin{\theta} & \cos{\theta} \end{bmatrix}, ........(1)$$ Physically, The matrix $$O$$ represents the rotational transformation, which describes the rotation in a 2D plane. This is well known among the researchers.

Now coming to less-popular Matrices/Group:

A pseudo-orthogonal matrix $$\cal{D}$$, which belongs to a pseudo-orthogonal group, is characterized as $$\cal{D}^T \eta \cal{D}=\eta ,$$ where $$\eta$$ is some constant metric.

Can anyone please give me a example of matrix $$\cal{D}$$ like in Eq.(1)??.

What physical transformation represents the matrix $$\cal{D}$$ (e.g. $$O$$ represents the rotation in 2D plane) ??.

• If you are not familiar with the Poincar\'e or the conformal group, you can check that their generators have representations as pseudo-orthogonal matrices. – G K Sep 10 '18 at 8:13
• I'm voting to close this question as off-topic because it shows insufficient prior research. – AccidentalFourierTransform Sep 10 '18 at 18:06
• @Sachin answers don't make questions on-topic – Kyle Kanos Sep 11 '18 at 10:06

A well-known physical use of an indefinite/pseudo orthogonal group $O(p,q)$ is the Lorentz group $O(n,1)$ in SR.
Orthogonal matrix $$O$$ (orthogonal group) which corresponds to a circular rotation. Similarly there exist a pseudo-orthogonal matrix $$\cal D$$ in pseudo orthogonal group, which corresponds to a hyperbolic rotation. $$\cal D$$ is as-
$$\cal D = \begin{bmatrix} \phantom{-} \cosh{\theta} & \sinh{\theta} \\ \sinh{\theta} & \cosh{\theta} \end{bmatrix},$$
which is pseudo-orthogonal as $$\cal D^t \eta \cal D=\eta$$, where $$\eta$$ = diag(1,-1).