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to be a matrix that diagonalizes A. with a diagonalizable coefficient
Solution by Diagonalization
The preceding discussion suggests the following procedure for solving a system
matrix A.
Step 1. Find a matrix P that diagonalizes A.
Step 2. Make the substitutions and to obtain a new “diagonal system” , where .
Step 3. Solve .
Step 4. Determine y from the equation .
EXAMPLE 2 Solution Using Diagonalization
(a) Solve the system
(b) Find the solution that satisfies the initial conditions , .
Solution (a)
The coefficient matrix for the system is
As discussed in Section 7.2, A will be diagonalized by any matrix P whose columns are linearly independent eigenvectors of
A. Since
the eigenvalues of A are , . By definition,
is an eigenvector of A corresponding to if and only if x is a nontrivial solution of —that is, of
