. . Application of matrices to Geometry There is a special type of non-singular matrices which are widely used in applications of matrices to geometry. For simplicity, we consider two-dimensional analytical geometry.
Let O be the origin, and x O x ' and y Oy ' be the x -axis and y -axis. Let P be a point in the plane whose coordinates are ( , ) x y with respect to the coordinate system. Suppose that we rotate the x -axis and y -axis about the origin, through an angle θ as shown in the figure. Let X OX ' and Y OY ' be the new X -axis and new Y -axis.
Let ( , ) X Y be the new set of coordinates of P with respect to the new coordinate system. Referring to Fig. . , we get x = OL ON LN X QT X Y θ , y = PL PT TL QN PT X Y θ .
These equations provide transformation of one coordinate system into another coordinate system. The above two equations can be written in the matrix form = cos X Y Let W = cos . Then x W X Y = and W = So, W has inverse and W − = − θ . We note that W W T − = .
Then, we get the inverse transformation by the equation X Y = W − = − = si Hence, we get the transformation X θ , Y θ . This transformation is used in Computer Graphics and determined by the matrix W = . We note that the matrix W satisfies a special property W W T − = ; that is, WW W W I T T Definition . A square matrix A is called orthogonal if AA A A I T T = .
Note A is orthogonal if and only if A is non-singular and A A T −− == q O X T L ′ x P Q Y ′ y ′ Y ′ X M N q Fig. . - - Applications of Matrices and Determinants Example . Prove that cos is orthogonal.
Let A = . Then, A T T = − θ . So, we get AA T = cos − cos sin cos sin = = I . Similarly, we get A A T = I .
Hence AA T = A A T = I ⇒ A is orthogonal. Example . If A is orthogonal, find a b , and c , and hence A − . If A is orthogonal, then AA A A T T = I .
So, we have AA T = I ⇒ ⇒ c + = ⇒ ⇒ ⇒ a So, we get A = and hence, A A T − = - -