Math 304
Linear Algebra
Harold P. Boas
June 28, 2006
Highlights
From last time:
Iapplication of eigenvectors to systems of differential
equations
Today:
Idiagonalization of matrices and applications
When is a matrix diagonal?
Suppose a linear operator Lon R3is represented in a basis
[u1,u2,u3]by the matrix
200
030
005
.
This means that Lu1=2u1and Lu2=3u2and Lu3=5u3.
In other words, the basis vectors are eigenvectors of the
operator L.
A square matrix Ais diagonalizable if the linear transformation
x7→ Axcan be represented in some basis by a diagonal matrix;
in other words, if there is a basis consisting of eigenvectors
of A; equivalently, if there is an invertible matrix Ssuch that
S−1AS is a diagonal matrix.
Example: exercise 1(b), p. 340
Diagonalize the matrix A=5 6
−2−2. In other words, find an
invertible matrix Sand a diagonal matrix Dsuch that
S−1AS =Dor A=SDS−1.
Solution. First find the eigenvalues and eigenvectors of A. The
vector 3
−2is an eigenvector with eigenvalue 1, and the
vector 2
−1is an eigenvector with eigenvalue 2. The matrix
S=3 2
−2−1is the transition matrix from the eigenvector
basis to the standard basis, and the matrix S−1AS is the
diagonal matrix 1 0
0 2.
