Mathematical Methods in the Physical Sciences · Problem 30

COORDINATE TRANSFORMATIONS; TENSOR ANALYSIS — Mary Boas

Mathematics — Methods · diagonalization-deformation

Problem

Let the matrix

M = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}

describe a deformation of the

(x, y)

plane. Find: the eigenvalues and eigenvectors of the transformation, the matrix

C

which diagonalizes

M

and specifies the rotation to new axes

(x', y')

along the eigenvectors, and the matrix

D

which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.