test 2 solve c the defination The definition of an eigenval

test 2 , solve c the defination

The definition of an eigenvalue and eigenvector of an n times n matrix A.

Solution

Denition:

A scalar is called an eigenvalue of the n × n matrix A if there is a nontrivial solution x of Ax = x.

Such an x is called an eigenvector corresponding to the eigenvalue .

Let’s nd an equivalent equation in standard form.
Ax = x
Ax x = 0
Ax Ix = 0
(A I)x = 0
Thus x is an eigenvector of A corresponding to the eigenvalue if and only if x and satisfy (AI)x =0.
Simply

Eigen value: The roots of the characteristic eqation |(AI)|=0

Eigen vector: The vector which satisfies the equation Ax=x