Application of matrices to Geometry

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.1.1, we get

x = OL = ON − LN = X cos θ – QT =  X cos θ − Y sin θ ,

y = PL = PT + TL = QN + PT =  X sin θ + Y cos θ .

These equations provide transformation of one coordinate system into another coordinate system.

The above two equations can be written in the matrix form

Hence, we get the transformation X = x cosθ - y sinθ , Y = x sinθ + y cosθ .

This transformation is used in Computer Graphics and determined by the matrix 

We note that the matrix W satisfies a special property W ^-1 = W^T ; that is, WW ^T = W^TW = I .

Definition 1.3

A square matrix A is called orthogonal if AA^T = A^TA = I.

Note

A is orthogonal if and only if A is non-singular and A^−1 = A^T .

Example 1.11

Prove that  is orthogonal.

Solution

Similarly, we get A^TA = I₂ . Hence AA^T = A^TA = I₂ ⇒ A is orthogonal.

Example 1.12

If A = , is orthogonal, find a, b and c , and hence A^−1.

Solution

If A is orthogonal, then AA^T = A^T A = I₃ . So, we have