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

Thornton & Marion, Classical Dynamics of Particles and Systems, 5th Edition

Chapter 1. Matrices, Vectors, and Vector Calculus

Problem 05. The determinant of the transformation matrix

The problem asks you to

  • show that $latex |\\bold{\\lambda}|^2 = 1$

This problem assumes

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

You should know about

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


Since the transformation matrix $latex \\bold{\\lambda}$ is

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

$latex |\\lambda| = \\lambda_{11}\\lambda_{22} – \\lambda_{12}\\lambda_{21}$


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

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

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


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

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