Eigenvectors

Eigenvectors

Definition. If A is an n x n matrix, then a nonzero vector x in Rⁿ is called an eigenvector of A if Ax is a scalar multiple of x; in other words,

for some scalar . The scalar is called an eigenvalue of A and x is the eigenvector corresponding to .

Theorem: If A is an n x n matrix, then the following are equivalent.

a) is an eigenvalue of A.

b) The system of equations (I - A)x = 0 has nontrivial solutions.

c) There is a nonzero vector x in Rⁿ such that Ax = x.

d) is a real solution of the characteristic equation det(I - A) = 0.

A.8a

Eigenvectors

Definition. If A is an n x n matrix, then a nonzero vector x in Rn is called an eigenvector of A if Ax is a scalar multiple of x; in other words,

for some scalar . The scalar is called an eigenvalue of A and x is the eigenvector corresponding to .

Theorem: If A is an n x n matrix, then the following are equivalent.

a) is an eigenvalue of A.

b) The system of equations (I - A)x = 0 has nontrivial solutions.

c) There is a nonzero vector x in Rn such that Ax = x.

d) is a real solution of the characteristic equation det(I - A) = 0.

Definition. If A is an n x n matrix, then a nonzero vector x in Rⁿ is called an eigenvector of A if Ax is a scalar multiple of x; in other words,

c) There is a nonzero vector x in Rⁿ such that Ax = x.