How to Prove a Rotation Matrix is Orthogonal

To prove a rotation matrix is orthogonal, you demonstrate that it satisfies the defining properties of an orthogonal matrix, specifically that its transpose equals its inverse ($R^T = R^{-1}$).

Understanding Orthogonal Matrices

An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors. A key property defining an orthogonal matrix $R$ is:

• $R^T R = I$, where $R^T$ is the transpose of $R$ and $I$ is the identity matrix.

This property is equivalent to saying that the transpose of the matrix is equal to its inverse ($R^T = R^{-1}$).

The Proof: Connecting Rotation Properties to Orthogonality

A rotation is a geometric transformation that moves points around a fixed point (like the origin in linear algebra) while preserving distances and angles. This preservation of geometric properties is key to proving its matrix representation is orthogonal.

Here's a common way to understand the proof:

1. Rotation Preserves Vector Lengths: A fundamental property of rotation is that it does not change the length (or magnitude) of a vector. If $v$ is any vector and $Rv$ is the vector after rotation by matrix $R$, then the length must remain the same: $|Rv| = |v|$.
2. Relating Length to the Dot Product: The square of a vector's length is equal to the dot product of the vector with itself: $|v|^2 = v \cdot v = v^T v$.
3. Applying Preservation to the Dot Product: Since rotation preserves length, the squared length is also preserved: $|Rv|^2 = |v|^2$.
4. Expanding the Rotated Dot Product: The dot product of the rotated vector with itself is $(Rv) \cdot (Rv)$. Using matrix notation and the property $(AB)^T = B^T A^T$, we get:
   $(Rv) \cdot (Rv) = (Rv)^T (Rv) = v^T R^T R v$.
5. Equating and Concluding: Since $|Rv|^2 = |v|^2$, we have $v^T R^T R v = v^T v$. Knowing $v^T v = v^T I v$ (where $I$ is the identity matrix), we get:
   $v^T R^T R v = v^T I v$.
   This equality must hold for any vector $v$. This can only be true if the matrices multiplying $v$ are equal: $R^T R = I$.

Thus, any matrix $R$ that represents a rotation must satisfy $R^T R = I$, which means it is an orthogonal matrix.

Rotation Matrices and Their Characteristics

As stated in the reference: "a square matrix R is a rotation matrix if and only if RT = R−1 and det R = 1".

This provides a concise characterization. The proof above shows why a matrix that performs a rotation necessarily satisfies the condition $R^T = R^{-1}$ (orthogonality). The additional condition, $\det R = 1$, distinguishes rotations from reflections, which are also orthogonal transformations but have a determinant of -1.

In summary, proving a rotation matrix is orthogonal involves showing that its geometric property of preserving lengths and angles translates directly into the algebraic property $R^T R = I$.