Question

Given that \( \lambda \) is an eigenvalue of matrix \( A \) with the corresponding eigenvector \( x \), and \( x \) is also an eigenvector of \( B = A - 2I \), find the relationship between \( \lambda \) and the eigenvalue of \( B \).

• A. \( \lambda + 2 \)
• B. \( \lambda \)
• C. \( 2\lambda \)
• D. \( \lambda - 2 \)

Hint:

When a matrix \( B \) is defined as \( B = A - cI \), where \( c \) is a constant and \( I \) is the identity matrix, the eigenvalues of \( B \) are the eigenvalues of \( A \) shifted by \( -c \).

Answer 1

The Correct Option is D

We are given that \( \lambda \) is an eigenvalue of \( A \) with corresponding eigenvector \( x \), i.e., \[ A \cdot x = \lambda \cdot x \] We are also given that \( x \) is an eigenvector of \( B = A - 2I \), where \( I \) is the identity matrix. Therefore: \[ B \cdot x = (A - 2I) \cdot x \] Substitute \( A \cdot x = \lambda \cdot x \) into the equation: \[ (A - 2I) \cdot x = \lambda \cdot x - 2 \cdot x \] Simplifying: \[ B \cdot x = (\lambda - 2) \cdot x \] Thus, the eigenvalue corresponding to matrix \( B \) is \( \lambda - 2 \).