For a given set of coordinate axes the components of a 2 nd rank tensor $T$ are given by $T_{i j}$.

(a) Show that if $\lambda$ is an eigenvalue of the matrix with elements $T_{i j}$ then it is also an eigenvalue of the matrix of the components of $T$ in any other coordinate frame.

Show that if $T$ is a symmetric tensor then the multiplicity of the eigenvalues of the matrix of components of $T$ is independent of coordinate frame.

A symmetric tensor $T$ in three dimensions has eigenvalues $\lambda, \lambda, \mu$, with $\mu \neq \lambda$.

Show that the components of $T$ can be written in the form

$$T_{i j}=\alpha \delta_{i j}+\beta n_{i} n_{j}$$

where $n_{i}$ are the components of a unit vector.

(b) The tensor $T$ is defined by

$$T_{i j}(\mathbf{y})=\int_{S} x_{i} x_{j} \exp \left(-c|\mathbf{y}-\mathbf{x}|^{2}\right) d A(\mathbf{x})$$

where $S$ is the surface of the unit sphere, $\mathbf{y}$ is the position vector of a point on $S$, and $c$ is a constant.

Deduce, with brief reasoning, that the components of $T$ can be written in the form (1) with $n_{i}=y_{i}$. [You may quote any results derived in part (a).]

Using suitable spherical polar coordinates evaluate $T_{k k}$ and $T_{i j} y_{i} y_{j}$.

Explain how to deduce the values of $\alpha$ and $\beta$ from $T_{k k}$ and $T_{i j} y_{i} y_{j}$. [You do not need to write out the detailed formulae for these quantities.]