Instructions: Read textbook pages 145 to 147 before working on the homework problems. Show all steps to get full credits.
- Let A be a matrix such that the entries in each row add up to 1. Show that the vector with all entries qual to 1 is an eigenvector. What is the corresponding eigenvalue?
- For an n × n matrix A prove that:
- If λ is an eigenvalue, u a corresponding eigenvector c a scalar, then λ+c is an eigenvalue of A+cI and u is a corresponding eigenvector.
- If the entries of each row of A add up to 0, then 0 is an eigenvalue, and thus the matrix is not invertible.
- Let Q be a unitary matrix and λ be an eigenvalue of Q, prove that |λ| =1.
