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Eigenvalue, Eigenvector

• Brief review
• Definition :
  • Eigenvector : Given a square matrix A, a non-zero vector v is called an eigenvector of A, if we can find a real number λ, such that
  • Eigenvalue : The number λ is the eigenvalue of A corresponding to the eigenvector v.

• Brief review
  • How to calculate the eigenvalue and eigenvector:

Use Characteristic Equation to calculate the

eigenvalue and then solve the linear system

with each λ and get the non-zero solutions, which

are thus eigenvectors.

• Brief review
  • Important Properties:

Give a square matrix , the eigenvalues satisfies:

Solution:

We get

Then all the eigenvectors corresponding to are

Solution:

4, when , solve the equation

We get

Then all the eigenvectors corresponding to are

Where k1 and k3 are not both zero.

Question 2: Suppose square matrix A have eigenvalue λ and

eigenvector v corresponding to λ, proof the following:

1, is the eigenvalue of

2, when A is invertible, is the eigenvalue of

Proof:

1,

iterate m-2 times and we get

2,

3,

Question 3: Suppose 3-order square matrix A have eigenvalue 1, 2,

3, then calculate

• Advanced topics:
  • Important Properties:

Supoose a square matrix A has the eigenvalues and the corresponding eigenvectors

If are different with each other, then is linear independent

• Advanced topics:
  • Proof: suppose there are coefficients such that

• Advanced topics:
  • Geometric Interpretation:

For : the matrix A can be viewed as a transformation of vectors v can be viewed as a vector λ can be viewed as a constant Then, means v is the vector such that after transformation of A, the result do not change direction and only stretch in length with λ

• Advanced topics:
  • Example of Geometric Interpretation:

Suppose , then A represents the transformation

which reverse the vertical coordinate only:

Then we can guess:

the vector on the X Axis are the eigenvetors of A:

the vector on the Y Axis are the eigenvetors of A:

Check by yourselves!