Question 7.2.6: Describe all possible cases for the number of real eigenvalu......
Describe all possible cases for the number of real eigenvalues (with their algebraic multiplicities) of a 3 × 3 matrix A.
Either the characteristic polynomial factors completely,
f_A(λ) = (λ_1 − λ)(λ_2 − λ)(λ_3 − λ),
or it has a quadratic factor without real zeros:
f_A(λ) = (λ_1 − λ)p(λ), where p(λ) ≠ 0 for all real λ.
In the first case, the λi could all be distinct, two of them could be equal, or they could all be equal. This leads to the following possibilities.
Examples for each case follow.
Case 1 (see Figure 3)
A=\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}, f_A(λ) = (1 − λ)(2 − λ)(3 − λ), Eigenvalues 1, 2, 3
Case 2 (see Figure 4)
A=\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}, f_A(λ) = (1 − λ)^2(2 − λ), Eigenvalues 1, 1, 2
Case 3 (see Figure 5)
A=I_3, f_A(λ) = (1 − λ)^3, Eigenvalues 1, 1, 1
Case 4 (see Figure 6)
A=\begin{bmatrix}1&0&0\\0&0&-1\\0&1&0\end{bmatrix}, f_A(λ) = (1 − λ)(λ^2 + 1), Eigenvalues 1
You can recognize an eigenvalue λ_0 whose algebraic multiplicity exceeds 1 on the graph of f_A(λ) by the fact that f_A(λ_0) = f^{\prime}_A(λ_0) = 0 (the derivative is zero, so that the tangent is horizontal). The verification of this observation is left as Exercise 37.
| case | No. of Distinct Eigenvalues |
Algebraic Multiplicities |
| 1 | 3 | 1 each |
| 2 | 2 | 2 and 1 |
| 3 | 1 | 3 |
| 4 | 1 | 1 |
