Linear Algebra - Sebastian Pancratz

B0 = {e1, e2, . . . , en} e1 = e [ψ]B0
⎛
d1
⎜ 0
⎜
⎝
0
d2
⎞
0
⎟
⎠
0 dn
,
d1 = ψ(e, e)
B0 d1, . . . , dp > 0 dp+1, . . . , dp+q < 0 di = 0
1
i > p + q 1 ≤ i ≤ p + q ei √|di| ei B
rank ψ = p + q
⎛
Ip
[ψ]B = ⎝ −Iq
⎞
0
⎠ .
0 0
ψ s(ψ) = p − q
p = 1
2
1
(r + s) q = 2 (r − s) (p, q)
ψ
⎛
Ip
⎝ −Iq
⎞
0
⎠
⎛
Ip ′
⎝ −Iq
0
0 0
′
⎞
⎠
0 0
p = p ′ q = q ′
B = {v1, . . . , vp, vp+1, . . . , vp+q, vp+q+1, . . . , vn}
⎛
Ip
[ψ]B = ⎝ −Iq
⎞
0
⎠ .
0 0
X = 〈v1, . . . , vp〉 Y = 〈vp+1, . . . , vn〉 ψ X
p V ψ
Q( p i=1 λivi) = p i=1 λ2i ≥ 0 p i=1 λivi = 0 ψ
X X ′ ≤ V dim X ′ = p ′ ψ X ′
X ′ ∩ Y = {0} ψ Y dim X ′ + dim Y ≤ n
p ′ = dim X ′ ≤ n − dim Y = p
N = 〈vp+1, . . . , vp+q〉 ψ N q
V ψ
p q
p ψ X
p ψ q N