Continuum Mechanics Rotation Matrix Problem

[lanew]

Homework Statement

The components of stress in the $x_i$ reference Cartesian system at a point of interested have been determined to be:

$\left[\begin{array}{ccc}<br /> 500 & 0 & 300 \\<br /> 0 & 700 & 0 \\<br /> 300 & 0 & -100<br /> \end{array}\right] \mathrm{MPa}$

Determine the principal values and directions of stress. Determine the rotation tensor transforming the components of stress from the principal components into components along the $x_i$ reference Cartesian system.

Homework Equations

$\mathbf{A} = \mathbf{R}^T \mathbf{V} \mathbf{R}$

where $\mathbf{A}$ is the original stress tensor, $\mathbf{R}$ is the rotation tensor, and $\mathbf{V}$ is a matrix of eigenvectors.

The Attempt at a Solution

I've solved for the principal values and directions, but don't know how to solve for the rotation tensor. It seems there's too many unknowns or I'm not making a necessary assumption. Does anyone have any suggestions?

Thank You.

 

[lanew]

I think I figured it out using diagonalization.

I combined the principal eigenvectors into a matrix $\mathbf{R}$ and checked by multiplying it by the diagonal matrix of the principal values.

$\mathbf{A}=\mathbf{R}\mathbf{\lambda}\mathbf{R}$

$\left[\begin{array}{ccc}<br /> 500 & 0 & 300 \\<br /> 0 & 700 & 0 \\<br /> 300 & 0 & -100<br /> \end{array}\right] = <br /> \left[\begin{array}{ccc}<br /> V_1 & V_2 & V_3 \\<br /> V_1 & V_2 & V_3 \\<br /> V_1 & V_2 & V_3<br /> \end{array}\right]<br /> \left[\begin{array}{ccc}<br /> \lambda_1 & 0 & 0 \\<br /> 0 & \lambda_2 & 0 \\<br /> 0 & 0 & \lambda_3<br /> \end{array}\right]<br /> \left[\begin{array}{ccc}<br /> V_1 & V_1 & V_1 \\<br /> V_2 & V_2 & V_2 \\<br /> V_3 & V_3 & V_3<br /> \end{array}\right]$

When I evaluated the right side, it equated to the left.

Does this seem correct?

Thanks Again.