Problem-19

eigenvalues
singular values

Question

Consider the matrix \(A = \begin{bmatrix}1 & 0\\C & 1\end{bmatrix}\), where \(C\) is some real number.

  1. What are the eigenvalues of \(A\)?

  2. Suppose \(\sigma_1\) and \(\sigma_2\) are the two singular values of \(A\), what is \(\sigma_1^2 + \sigma_2^2\)?

Solution

Since \(A\) is a triangular matrix, the eigenvalues are the elements on the diagonal. \(1\) is the only eigenvalue here, but repeated twice. The singular values of \(A\) are the square roots of the eigenvalues of \(A^TA\). \[ A^TA = \begin{bmatrix} 1 & C\\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0\\ C & 1 \end{bmatrix} = \begin{bmatrix} 1 + C^2 & C\\ C & 1 \end{bmatrix} \] We are asked to find \(\sigma_1^2 + \sigma_2^2\). This is the sum of the roots of the following characteristic polynomial:

\[ \begin{aligned} \left[(1 + C^2) - \lambda\right](1 - \lambda) - C^2 &= 0\\ \implies \lambda^2 - (2 + C^2) \lambda + (1 - C^2) &= 0 \end{aligned} \] The required sum is therefore \(\boxed{2 + C^2}\).