Find spectral theorem of $A$ and find its singular eigenvalues.

Question

The rotation matrix $$A=\pmatrix{ \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta}$$ has complex eigenvalues $\{e^{\pm i\theta}\}$ corresponding to eigenvectors $\pmatrix{1 \\i}$ and $\pmatrix{1 \\ -i}$.

Now I want to find $A = \lambda_1v_1v_1^T + \lambda_2v_2v_2^T$. and then I want to find singular values of $A$ (which I believe only exist in SVD)

Step 1: make $v_1,v_2$ orthonormal vectors.

Step 2: Calculate $A$, by the spectrom theorem I wrote above.

Step 3: Calculate $A^TA$ because we want to find singular values of $A$.

Step 4: How do I find singular values now of $A$? do I need to use the eigenvectors I found above? $A^TAv = \lambda_{1,2}v$?

Any help is highly appreciated.

Answer 1

Normally the singular values of a matrix $A$ are defined as the (positive) square roots of the eigenvalues of $A^*A$.