Linear Systems of First-Order Differential Equations 361
THE MATRIX WHOSE EIGE!VVALUES AND EIGENVECTORS ARE TO BE CALCULATED IS
l.OOOOEtOO -1. OOOOEtOO l. OOOOEtOO
-1. OOOOEtOO l.OOOOEtOO l. OOOOEtOO
l. OOOOEtOO l.OOOOEtOO l. OOOOEtOO
AN EIGENVALUE I S 2. 00000 EtOO + O.OOOOOEtOO I
THE ASSOCIATED EIGENVECTOR I S
- 94 42 7E-Ol + O.OOOOOEtOO I
-4. 4 72 14E-Ol + O.OOOOOEtOO I
- 4 72 14E-Ol + O.OOOOOEtOO I
AN EIGENVALUE I S -1.^00000 E+OO + O.OOOOOEt OO I
THE ASSOCIATED EIGE!VVECTOR I S
- 4 53 56E-Ol + O.OOOOOEtOO I
- 4 5356 E-Ol + O. OOOOOEtOO I
- 45356 E-Ol + O.OOOOOEtOO I
AN EIGENVALUE I S 2. 00 000 EtOO + O. OOOOOEtOO I
THE ASSOCIATED EIGENVECTOR I S
- 11803 E-Ol + O.OOOOOEtOO I
- 440 98E- Ol + O.OOOOOEtOO I
6.5 5902 E-Ol + O.OOOOOEtOO I
F igure 8.3 Eigenvalues and Eigenvectors for Example 5.3.
- In example 4, we manu ally calculated the eigenvalues of t he given
matrix to be .A 1 = 2 + i and .A 2 = 2 - i. And we found the associ-
ated eigenvectors to be
X1 - (-1 - l + i)
(
and x2 = - 1 - i)
1.
When the entries of a matrix are all real, complex eigenvalues and their
associated eigenvectors occur in complex conjugate pairs. Observe in
this instance .A 1 and .A 2 are complex conjugate scalars and x 1 and x 2
are complex conjugate vectors. Results for the given matrix computed
using EIGEN are shown in Figure 8.4. The eigenvalues computed are
identical with those computed by hand, but the associated eigenvectors
are
z1 = (-\-i) and z 2 = (-~;i).
Notice that z 1 and z 2 are complex conjugate vectors. Also notice z 1 =
i x 1 and z2 = -ix2.
