The orthogonal group Let On denote the set of all n x n orth

The orthogonal group. Let O(n) denote the set of all n x n orthogonal matrices. (1 pt) Show that the inverse matrix of an orthogonal matrix is an orthogonal matrix too. Show that the product AB of two orthogonal matrices A, B belongsto O(n) is again an othogonal matrix. Remark. These two facts show that the set of orthogonal matrices O(n) is a group, called the orthogonal group.

Solution

We begin the answer by narrating the definition and certain properties of an orthogonal matrix. An orthogonal matrix is a square matrix with real entries whose columns and rows are orthogonal unit vectors (i.e., orthonormal vectors), i.e. Q^TQ = QQ^T = I where I is the identity matrix. This leads to the equivalent characterization i.e. a matrix Q is orthogonal if its transpose is equal to its inverse i.e. Q^T = Q^-1. An orthogonal matrix Q is necessarily invertible (with inverse Q¹ = Q^T). An orthogonal matrix must be symmetric.

( i ) Let Q be an orthogonal matrix. Then Q^-1 = Q^T = Q ( as an orthogonal matrix is necessarily symmetric) . This implies that Q^-1 is orthogonal. ( Even otherwise as the transpose of Q^T is Q which is equal to Q^T, therefore, ( Q^T)^T = Q = Q^T , so that Q^T is orthogonal. Hence Q^-1 = Q^T is orthogonal.

(ii) If A and B are two n x n orthogonal matrices, then (AB)^T(AB)= (B^TA^T)(AB)= B^T(A^TA)B= B^T I B = B^T B = I ( as both A and B are orthogonal so that A^TA = I and B^TB = I). Thus AB is orthogonal.