Solution Found!
Let J be the n × n matrix of all 1’s, and consider Use the
Chapter , Problem 15E(choose chapter or problem)
Let J be the \(n \times n\) matrix of all 1’s, and consider \(A=(a-b) I+b J\); that is,
\(A=\left[\begin{array}{ccccc}a & b & b & \cdots & b \\ b & a & b & \cdots & b \\ b & b & a & \cdots & b \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b & b & b & \cdots & a\end{array}\right]\)
Use the results of Exercise 16 in the Supplementary Exercises for Chapter 3 to show that the eigenvalues of A are a - b and a + (n - 1)b. What are the multiplicities of these eigenvalues?
Questions & Answers
QUESTION:
Let J be the \(n \times n\) matrix of all 1’s, and consider \(A=(a-b) I+b J\); that is,
\(A=\left[\begin{array}{ccccc}a & b & b & \cdots & b \\ b & a & b & \cdots & b \\ b & b & a & \cdots & b \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b & b & b & \cdots & a\end{array}\right]\)
Use the results of Exercise 16 in the Supplementary Exercises for Chapter 3 to show that the eigenvalues of A are a - b and a + (n - 1)b. What are the multiplicities of these eigenvalues?
ANSWER:Solution 15E
The eigenvalues of the matrix are