Orthogonal vs orthonormal matrices – what are simplest possible definitions and examples of each ?

matricesorthogonalityorthonormal

I'm trying to understand orthogonal and orthonormal matrices and I'm very confused. Unfortunately most sources I've found have unclear definitions, and many have conflicting definitions!

Some sites like for example https://en.wikipedia.org/wiki/Orthogonal_matrix seem to imply these are the same thing, but most others at least imply if not directly state that they are different.

Before someone marks this question as a duplicate I've already consulted Difference between orthogonal and orthonormal matrices and the provided answers do not make the definitions of orthogonal and orthonormal matrices clear, at least not to me.

Also, I've been able to find various definitions and verbiage, but I've been able to find very few, if any, examples of matrices that are or aren't orthogonal/orthonormal/both/neither.

Additionally I should clarify I'm concerned with the definition of orthogonal and orthonormal matrices, not orthogonal/orthonormal vectors.

Based on the sources available and my current understanding of them, this seems to be the definition of each:

def. of orthogonal matrix:
$AA^T = I$

and:
$A^{-1} = A^T$

def. of orthonormal matrix:
meets the definition of orthogonal (above) and also:
$determinant(A) = 1$

So my questions are:

  1. Are these definitions correct? Please don't throw complicated math formulas at me here, I'm ideally looking for simple logic that I could code in a Python/NumPy if statement (I'm an engineer not a math person!). If these definitions are not correct, what should they be?

  2. What would be examples of matrices that are:
    a) not orthogonal or orthonormal
    b) orthogonal but not orthonormal
    c) orthonormal but not orthogonal (if this is possible, my current understanding is it's not)
    d) orthogonal and orthonormal

Best Answer

(Perhaps slightly confusingly), orthogonal matrices are those whose columns and rows are orthonormal. Equivalently, a matrix $\mathbf A$ is orthogonal if $$\mathbf A \mathbf A^{T} = \mathbf A^{T} \mathbf A = \mathbf I,$$ where $\mathbf I$ is the identity matrix.

I've not heard the term orthonormal matrix before. Generally, those matrices that are both orthogonal and have determinant $1$ are referred to as special orthogonal matrices or rotation matrices. If I read "orthonormal matrix" somewhere, I would assume it meant the same thing as orthogonal matrix.

Some examples:

  1. $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not orthogonal.
  2. $\begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}$ is orthogonal, but has determinant $-1$ so is not special orthogonal.
  3. $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is also a matrix that has determinant $1$ but is not orthogonal. I agree that if "orthonormal" means special orthogonal, then a matrix cannot be orthonormal without also being orthogonal.
  4. $\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is orthogonal and has determinant $1$, so is special orthogonal.
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