Abstract Algebra Theory and Applications - Computer Science ...

10.1 MATRIX GROUPS 177we know that 〈x, (A t A − I)x〉 = 0 for all x. Therefore, A t A − I = 0 orA −1 = A t .(3) ⇒ (4). If A is inner product-preserving, then A is distance-preserving,since‖Ax − Ay‖ 2 = ‖A(x − y)‖ 2= 〈A(x − y), A(x − y)〉= 〈x − y, x − y〉= ‖x − y‖ 2 .(4) ⇒ (5). If A is distance-preserving, then A is length-preserving.Letting y = 0, we have‖Ax‖ = ‖Ax − Ay‖ = ‖x − y‖ = ‖x‖.(5) ⇒ (3). We use the following identity to show that length-preservingimplies inner product-preserving:〈x, y〉 = 1 2[‖x + y‖ 2 − ‖x‖ 2 − ‖y‖ 2] .Observe that〈Ax, Ay〉 = 1 [‖Ax + Ay‖ 2 − ‖Ax‖ 2 − ‖Ay‖ 2]2= 1 [‖A(x + y)‖ 2 − ‖Ax‖ 2 − ‖Ay‖ 2]2= 1 [‖x + y‖ 2 − ‖x‖ 2 − ‖y‖ 2]2= 〈x, y〉.Example 6. Let us examine the orthogonal group on R 2 a bit more closely.An element T ∈ O(2) is determined by its action on e 1 = (1, 0) t and e 2 =(0, 1) t . If T (e 1 ) = (a, b) t , then a 2 + b 2 = 1 and T (e 2 ) = (−b, a) t . Hence, Tcan be represented by( ) ( )a −b cos θ − sin θA ==,b a sin θ cos θwhere 0 ≤ θ < 2π. A matrix T in O(2) either reflects or rotates a vector inR 2 (Figure 10.2). A reflection is given by the matrix( ) 1 0,0 −1□