Page 669 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

If , this system becomes
Solving this system yields , , so
Thus
is a basis for the eigenspace corresponding to . Similarly, the reader can show that

is a basis for the eigenspace corresponding to . Thus

diagonalizes A, and

Therefore, the substitution
yields the new “diagonal system”

From 2 the solution of this system is

so the equation yields, as the solution for y,

or

(8)

Solution (b)

If we substitute the given initial conditions in 8, we obtain

Solving this system, we obtain , , so from 8, the solution satisfying the initial conditions is