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:

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

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}

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

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

Does this seem correct?

Thanks Again.