11

13A

Domain Range

4

1

5

1

4

1

11

13

5

1

11

13A

Domain Range

1

1

0

4

1

1

11

13

0

4

11

13A

Domain Range

1

1

2

2

1

1

11

13

2

2

11

13A

Domain Range

definition: T is a linear transformation ,

vvT

v

0

EIGENVECTOR

EIGENVALUE

1

2;

1

3

73

188

vu

AA is the matrix for a linear transformation Trelative to the STANDARD BASIS

1

20

1

32

2

6

1

3

73

188uA

1

21

1

30

1

2

1

2

73

188vA

1

2,

1

3

1

20

1

32

2

6

1

3

73

188uA

1

21

1

30

1

2

1

2

73

188vA

1

2,

1

3

1

20

1

32

2

6

1

3

73

188uA

1

21

1

30

1

2

1

2

73

188vA T

T

1

2,

1

3

1

20

1

32

2

6

1

3

73

188uA

1

21

1

30

1

2

1

2

73

188vA T

T

The matrix for T relative to the basis

10

02

1

2,

1

3

1

20

1

32

2

6

1

3

73

188uA

1

21

1

30

1

2

1

2

73

188vA T

T

The matrix for T relative to the basis

10

02

Eigenvectors for T

Diagonal matrix

The matrix for a linear transformation T

relative to

a basis of eigenvectors

will be diagonal

To find eigenvalues and eigenvectors for a given matrix A:

Solve for and v

A v v=

A v v= I

A vv= I0 -

A ) v= I0 -(

To find eigenvalues and eigenvectors for a given matrix A:

Solve for and

A ) v= I0 -(

Remember: 0v

v is a NONZERO vector in the null space of the matrix:

A )I -(

v is a NONZERO vector in the null space of the matrix:

A )I -(

The matrix has a nonzero vector in its null space iff:A )I -(

A )I -(det = 0

A )I -(det = 0

73

188A

A = I -

0

0

73

188-

73

188

A )I -(det =

122

547873

188

2

det

A )I -(det =

122

547873

188

2

det

A )I -(det =

122

547873

188

2

det

This is called the characteristic polynomial

A )I -(det =

122

547873

188

2

det

= 0

the eigenvalues are 2 and -1

A )I -( =

93

186

723

1882

the null space of 2I - A =

2

1

3

the eigenvectors belonging to 2 are nonzero vectors in the null space of 2I - A

A )I -( =

63

189

713

1881

the null space of -1I - A =

-1

1

2

the eigenvectors belonging to -1 are nonzero vectors in the null space of -1I - A

PAPB

andrseigenvecto

andseigenvalueA

1

1

11

23

73

188

11

23

10

02

1

2

1

3:

12:73

188

PAPB

andrseigenvecto

andseigenvalueA

1

1

11

23

73

188

11

23

10

02

1

2

1

3:

12:73

188

Matrix for T relative to standard basis

PAPB

andrseigenvecto

andseigenvalueA

1

1

11

23

73

188

11

23

10

02

1

2

1

3:

12:73

188

Matrix for T relative to columns of P

PAPB

andrseigenvecto

andseigenvalueA

1

1

11

23

73

188

11

23

10

02

1

2

1

3:

12:73

188

Basis of eigenvectors

PAPB

andrseigenvecto

andseigenvalueA

1

1

11

23

73

188

11

23

10

02

1

2

1

3:

12:73

188

eigenvalues