A First Course in Linear Algebra, 2017a

7.2. Diagonalization 363
7.2.3 Complex Eigenvalues
In some applications, a matrix may have eigenvalues which are complex numbers. For example, this often
occurs in differential equations. These questions are approached in the same way as above.
Consider the following example.
Example 7.26: A Real Matrix with Complex Eigenvalues
Let
⎡
1 0
⎤
0
A = ⎣ 0 2 −1 ⎦
0 1 2
Find the eigenvalues and eigenvectors of A.
Solution. We will first find the eigenvalues as usual by solving the following equation.
⎛
⎡
det⎝x⎣
1 0 0
0 1 0
0 0 1
⎤
⎡
⎦ − ⎣
1 0 0
0 2 −1
0 1 2
⎤⎞
⎦⎠ = 0
This reduces to (x − 1) ( x 2 − 4x + 5 ) = 0. The solutions are λ 1 = 1,λ 2 = 2 + i and λ 3 = 2 − i.
There is nothing new about finding the eigenvectors for λ 1 = 1sothisisleftasanexercise.
Consider now the eigenvalue λ 2 = 2 + i. As usual, we solve the equation (λI − A)X = 0asgivenby
⎛ ⎡ ⎤ ⎡ ⎤⎞
⎡ ⎤
1 0 0 1 0 0
0
⎝(2 + i) ⎣ 0 1 0 ⎦ − ⎣ 0 2 −1 ⎦⎠X = ⎣ 0 ⎦
0 0 1 0 1 2
0
In other words, we need to solve the system represented by the augmented matrix
⎡
1 + i 0 0
⎤
0
⎣ 0 i 1 0 ⎦
0 −1 i 0
We now use our row operations to solve the system. Divide the first row by (1 + i) andthentake−i
times the second row and add to the third row. This yields
⎡
1 0 0
⎤
0
⎣ 0 i 1 0 ⎦
0 0 0 0
Now multiply the second row by −i to obtain the reduced row-echelon form, given by
⎡
1 0 0
⎤
0
⎣ 0 1 −i 0 ⎦
0 0 0 0