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 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
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 -( =
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
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