Orthogonal Transformations:
Definition.: Q is Orthogonal if 1/Q = Qt where Qt is the transpose of Q i.e. Q*Qt = Qt*Q = I.
Properties of Orthogonal Matrices:
1. An orthogonal matrix does not change the length of a vector in the L2 norm.
Proof: ||Qx||**2 = <Qx,Qx> = <x,Qt*Q x> = <x,x> = ||x||**2
2. A product of any number of orthogonal matrices is orthogonal as is the inverse and transpose of an orthogonal matrix.
Proof: Let P = Q1*Q2...*Qn. Pt = (Q1*Q2*..Qn)t = Qnt*...*Q2t*Q1t = 1/Qn*...*1/Q2*1/Q1. Pt*P = 1/Qn*...*1/Q2*1/Q1 * Q1*Q2...*Qn = I
3. A similarity transform of a symmetric matrix by an orthogonal matrix will still be symmetric: B=1/Q*A*Q is symmetric if A is and Q is orthogonal.
Proof: Bt = (1/Q*A*Q)t = Qt*At*(1/Q)t = 1/Q*A*Q = B.
4. If an orthogonal matrix is also symmetric then its square is the identity: Q*Q = Qt*Q = I
Examples of Orthogonal Matrices:
The most common examples of orthogonal matrices are rotations and reflections. Both are important for developing numerical methods.
The fact that orthogonal matrices don't change the lengths of vectors make them very desirable in numerical applications since they will not increase rounding errors significantly. They are therefore favored for stable algorithms.
They are particularly important for eigenvalue methods for symmetric matrices because they produce similarity transforms that preserve the symmetric property.