Linear Algebra, 2020a

Section IV. Jordan Form 451
and to move from the upper right to the lower right we multiply by this matrix.
( ) −1 ( )
1 −2 1/2 1/2
P =
=
1 2 −1/4 1/4
So this equation shows the similarity.
( )(
1/2 1/2 2 −1
−1/4 1/4 1 4
)(
)
1 −2
1 2
(
=
3 0
1 3
)
2.5 Example This matrix
⎛
⎞
4 1 0 −1
0 3 0 1
T = ⎜
⎟
⎝0 0 4 0 ⎠
1 0 0 5
has characteristic polynomial (x − 4) 4 and so has the single eigenvalue 4.
power p (T − 4I) p N ((T − 4I) p )
⎛
⎞ ⎛ ⎞
0 1 0 −1 −w
1
0 −1 0 1
⎜
⎟ {
w
⎜ ⎟ | z, w ∈ C}
⎝0 0 0 0 ⎠ ⎝ z ⎠
1 0 0 1 w
⎛
⎞ ⎛ ⎞
−1 −1 0 0 −y
2
1 1 0 0
⎜
⎟
⎝ 0 0 0 0⎠
{ y
⎜ ⎟ | y, z, w ∈ C}
⎝ z ⎠
1 1 0 0 w
⎛
⎞ ⎛ ⎞
0 0 0 0 x
3
0 0 0 0
⎜
⎟ {
y
⎜ ⎟ | x, y, z, w ∈ C}
⎝0 0 0 0⎠
⎝ z ⎠
0 0 0 0 w
The null space of t−4 has dimension two, the null space of (t−4) 2 has dimension
three, and the null space of (t − 4) 3 has dimension four. This gives the canonical
form for t − 4.
⎛
⎞
0 0 0 0
1 0 0 0
N = Rep B,B (t − 4) = ⎜
⎟
⎝0 1 0 0⎠
0 0 0 0
⃗β 1
t−4
↦−−→ ⃗β 2
t−4
t−4
↦−−→ ⃗β 3 ↦−−→ ⃗0
t−4
⃗β 4 ↦−−→ ⃗0