Thornton & Marion (5th Edition), Chapter 01, Problem 05 Solution

Thornton & Marion, Classical Dynamics, 5th Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 05. The determinant of the transformation matrix

The problem asks you to

  • show that |\bold{\lambda}|^2 = 1

This problem assumes

  • The transformation matrix \bold{\lambda} to be a two-dimensional orthogonal matrix.

You should know about

  1. Calculation of a determinant
  2. Properties of the orthogonal transformation matrix

Solution

Since the transformation matrix \bold{\lambda} is

\bold{\lambda} = \left( \begin{array}{cc} \lambda_{11} & \lambda_{12} \\ \lambda_{21} & \lambda_{22} \end{array} \right)
then determinant of this matrix is

|\lambda| = \lambda_{11}\lambda_{22} - \lambda_{12}\lambda_{21}
 

So

|\lambda|^2 = (\lambda_{11}\lambda_{22} - \lambda_{12}\lambda_{21})^2 = \lambda_{11}^2\lambda_{22}^2 + \lambda_{12}^2\lambda_{21}^2 - 2\lambda_{11}\lambda_{22}\lambda_{12}\lambda_{21}
 

From Equation (1.13), which is the orthogonality condition,

\sum_j = \lambda_{ij}\lambda_{kj} = \delta_{ik}
|\lambda|^2 = \lambda_{11}^2\lambda_{22}^2 = 1

Therefore,

|\bold{\lambda}|^2 = 1

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