Math 300
Notes for Section 7.1
1. (Eigenvalues and Eigenvectors of a Matrix). Let Abe an n๎nmatrix. A scalar ๎is called an
eigenvalue of Aif there isa nonzero vector Evin Rnsuch that
AEvD๎Ev:
The vector Evis called an eigenvector of Acorresponding to ๎. (Note that ๎may be 0.)
2. (Finding the Eigenvalues of A). Let Abe an n๎nmatrix. To find the eigenvalues of A, compute
the determinant det.๎I ๎A/ and solve the equation
det.๎I ๎A/ D0:
The real solutions of the equation are the eigenvalues of A. When expanded, the determinant
det.๎I ๎A/ is a polynomial in ๎of degree n, called the characteristic polynomial of A. The
equation det.๎I ๎A/ D0is called the characteristic equation of A.
3. (Finding the Eigenvectors Corresponding to ๎). Let ๎be an eigenvalue of a matrix A. To find
the eigenvectors corresponding to ๎, form the matrix ๎I ๎Aand solve the homogeneous system
.๎I ๎A/ExDE
0:
The nonzero solutions are the eigenvectors corresponding to ๎. The set of all solutions(the eigen-
vectors plus the zero vector) is called the eigenspace of ๎. The eigenspace, as the solution space
of a homogeneous system, is a subspace of Rn.