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The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose

Let's show that this definition is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.

Note that:

x^{T} y = (O x)^{T} (O y) = x^{T} O^{T} O y

and for that to be true for all possible

x

and

y

then we must have:

O^{T} O = I

i.e. the matrix inverse is equal to the transpose.

Conversely, if:

O^{T} O = I

is true, then

(O x)^{T} (O y) = x^{T} (O^{T} O) y = x^{T} y

These matricese are called the orthogonal matrices.

TODO is there any more intuitive way to think about this?

Table of contents 82 1
- Elements of the orthogonal group have determinant plus or minus one The orthogonal group is the group of all invertible matrices where the inverse is equal to the transpose

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Elements of the orthogonal group have determinant plus or minus one