Let $a$ and $b$ be distinct nonzero real numbers and let $A\in \mathbb{R}^{n\times n}(n\geq 2)$ with each diagonal entry equal to $a$ and each off-diagonal entry equal to $b$.
- Determine all eigenvalues and eigenvectors of $A$ together with their algebraic multiplicities.
- Is $A$ diagonalizable? Why?
- Determine the minimal polynomial of $A$.
My idea: Considering a $4\times 4$ matrix for simplicity:
$\left( \begin{array}{ccccc} a& b&b & b \\ b &a&b &b \\ b & b&a& b\\ b&b&b&a \end{array} \right)$ now we perform row operations to transform this matrix into a simple one like this: $\left( \begin{array}{ccccc} a-b& b-a& 0 & 0 \\ 0 &a-b& b-a&0 \\ 0 & 0&a-b & b-a\\ b&b&b&a \end{array} \right)$ Then find the eigen values and from there eigen vectors? I think eigen values are $a\pm b$ but not sure how to prove it.
Thanks Marvis, your answer is very helpful.
Can you also comment on my ideas about the rest of the problem: to find eigen vectors we need $x$ such that $(A-\lambda I)x=0.$ So, $\lambda =a-b$ gives me that $(A-\lambda I)=(b)$ a matrix with all entries $b$, thus eigen vector in this case is a $n\times 1$ vector with first entry $n-1$ and the rest of the entries as 1's. So, using the eigen vectors I can find an invertible matrix $P$ and find out if $P^{-1}AP=D$ where $D$ is a diagonal matrix? Then I can get minimal polynomial from the diagonal matrix.