Orthogonal matrix is a square matrix R=[xij] such that RT = R-1. In other words, a square matrix (R) whose transpose is equal to its inverse is known as orthogonal matrix i.e. RT = R-1.
Orthogonal matrix examples
The best example of an orthogonal matrix is an identity matrix or unit matrix as shown below.
\[I=\begin{bmatrix} 1 &0 \\ 0& 1 \end{bmatrix}\]
The example of an orthogonal matrix of order 3 (or matrix size is 3 x 3) is given as
\[I=\begin{bmatrix} 1 &0 &0 \\ 0& 1 &0 \\ 0& 0 & 1 \end{bmatrix}\]
Orthogonal matrix properties
1. Let B be an orthogonal matrix then BBT = BTB = I, where I is the identity matrix.
Orthogonal matrix determinant
The determinant of an orthogonal matrix is always has a modulus of 1.
Proof:
Let B be an orthogonal matrix then BBT = I (according to the property)
BBT = I
⇒ |BBT| = |I| = 1
⇒ |B| |BT| = 1
⇒ (|B|)2 = 1
⇒ |B| = ±1
Hence proved.
