# Orthogonal matrix   - properties and formulas -

- Definition
- Example
- Product
- Inverse
- Determinant
Definition
Let $R$ be a square matrix. $R$ is called orthtonal matrix if it satisfies
, where $R^T$ denotes the transpose of $R$, and $I$ denotes the identity matrix.
Example
Show that the following matrix
is an orthogonal matrix.
Explanation
By calculating concretely, we have
Therefore, $R$ is an orthogonal matrix.

Product
If $R$ and $S$ are orthogonal matrices, then the product $RS$ are an orthogonal matrix as well. It satisfies

Proof
Let $R$ and $S$ are orthogonal matrices. They satisfy
where $I$ is the identity matrix. By the property of product of transopsed matrix
We see that
and
Therefore we obtain
The product $RS$ are an orthogonal matrix as well.

Inverse
The inverse matrix of an orthogonal matrix R is the transpose of $R$,

Proof
In general, the inverse matrix of a square matrix $A$ is a matrix $B$ that satisfies
, where $I$ is the identity matrix. Such $B$ is expressed as
With this in mind, looking at the definition of an orthogonal matrix,
we see that $R^{T}$ is the inverse of $R$.

Determinant
The determinant of an orthogonal matrix $R$ is $\pm 1$.

Proof
Let $R$ be an orthogonal matrix. We have
The determinant is
$$\tag{5.1}$$ In the left-hand side, the property of product of determinant
and the property of determinant of transpose
give
In the right-hand side of $(5.1)$, the determinant of identity matrix is $1$. We have
Therefor we obtain