Documents Live, a web authoring and publishing system
Documents Live, a web authoring and publishing system
If you see this, something is wrong
First published on Saturday, Jan 11, 2025 and last modified on Saturday, Jan 11, 2025
Mathedu
Diagonalisation, Rotation
In that test, you will solve in three different ways the following linear system, whose coefficient matrix is the rotation matrix of angle \( \theta\) in an orthonormal basis.
We will assume that \( \sin(\theta)\ne 0\) , because otherwise the system is trivial, as \( \cos(\theta)=\pm 1\) .
Use the fact that \( \sin(\theta)\ne 0\) .
Mutliply the two members of the resulting equation by \( \sin(\theta)\) and use the fact that \( \sin^2(\theta)+\cos^2(\theta)=1\) .
Use the fact that \( \sin^2(\theta)+\cos^2(\theta)=1\) to simplify \( -1+\cos^2(\theta)\) , and distribute \( \frac{1}{\sin(\theta)}\) .
Specify the rotation matrix used as the coefficient matrix.
Use the complex exponential notation.
Don’t use the complex exponential notations for that question. For the first eignevalue, solve the first equation first, and check that the corresponding second equation is fulfilled. For the second eigenvalue, use the complex conjugate of the eigenvectors corresponding to the first eigenvalue.
Use the complex exponential notation for the eigenvalues.
Denote \( Y=\begin{bmatrix}z\\ t\end{bmatrix}\) and \( C=\begin{bmatrix}g\\ h\end{bmatrix}B\) . Use the fact that the invese of \( e^{i\theta}\) is \( e^{-i\theta}\) and reversely.
Use the fact that \( \frac{1}{i}=-i\) for the inverse of \( P\) .
Set \( \frac{1}{2}\) in factor and devlop it at the end. Use the fact that \( e^{i\theta}+e^{-i\theta}=2\cos(\theta)\) and \( e^{i\theta}-e^{-i\theta}=2i\sin(\theta)\) .
The 3 following methods to solve a liear system whose matrix of coefficients is a matrix of rotation:
elimination/substitution,
solving the equivalent matix equation by matrix inversion,
diagonalizing the coefficient matrix in \( \mathbb{C}\) ,
give the same solution.
And we learned to diagonalize a real matirx in \( \mathbb{C}\) .