Question-21

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

Part-(a)

What are the eigenvalues of \(A\)?

• \(1\) and \(0\)
• Only \(0\)
• Only \(1\)
• \(A\) does not have any real eigenvalues.

Answer

• \(1\) and \(0\)
• Only \(0\)
• Only \(1\)
• \(A\) does not have any real eigenvalues.

Solution

Since \(A\) is a lower triangular matrix, its eigenvalues are the elements on the diagonal. So \(1\) is the only eigenvalue.

Part-(b)

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

• \(0\)
• \(2\)
• \(C\)
• \(2 + C^2\)
• Insufficient data

Answer

• \(0\)
• \(2\)
• \(C\)
• \(2 + C^2\)
• Insufficient data

Solution

The squares of the singular values of \(A\) are the eigenvalues of \(A^TA\). \[ A^TA = \begin{bmatrix} 1 + C^2 & C\\ C & 1 \end{bmatrix} \] The characterstic polynomial of \(A^TA\) is: \[ \lambda^2 - \left( 2 + C^2 \right)\lambda + 1 = 0 \] The sum of the roots are \(\sigma_1^2 + \sigma_2^2\). This is given by \(2 + C^2\).