Characteristic polynomial

1. CHARACTERISTIC POLYNOMIAL Hanpen Robot

2. 𝑓: 𝑉 → 𝑉, dim 𝑉 = 2 𝑉 is linear space 𝑓 is linear map

3. Let 𝕧1and 𝕧2 be a basis on 𝑉. Then, 𝑓 𝕧1 = 𝑎11 𝕧1 + 𝑎12 𝕧2 𝑓 𝕧2 = 𝑎21 𝕧1 + 𝑎22 𝕧2

4. Let 𝕧1and 𝕧2 be a basis on 𝑉. Then, 𝑓 𝕧1 = 𝑎11 𝕧1 + 𝑎12 𝕧2 𝑓 𝕧2 = 𝑎21 𝕧1 + 𝑎22 𝕧2

5. 𝑓 0 0 𝑓 𝕧1 𝕧2 = 𝑎11 𝑎12 𝑎21 𝑎22 𝕧1 𝕧2 ⇓ 𝑓 − 𝑎11 −𝑎12 −𝑎21 𝑓 − 𝑎22 𝕧1 𝕧2 = 𝕠 𝕠

6. 𝑓 0 0 𝑓 𝕧1 𝕧2 = 𝑎11 𝑎12 𝑎21 𝑎22 𝕧1 𝕧2 ⇓ 𝑓 − 𝑎11 −𝑎12 −𝑎21 𝑓 − 𝑎22 𝕧1 𝕧2 = 𝕠 𝕠

7. 𝑓 − 𝑎22 𝑎12 𝑎21 𝑓 − 𝑎11 𝑓 − 𝑎11 −𝑎12 −𝑎21 𝑓 − 𝑎22 𝕧1 𝕧2 = 𝕠 𝕠 Where 𝑓 − 𝑎22 𝑎12 𝑎21 𝑓 − 𝑎11 is cofactor matrix.

8. 𝑓 − 𝑎22 𝑓 − 𝑎11 − 𝑎12 𝑎12 0 0 𝑓 − 𝑎22 𝑓 − 𝑎11 − 𝑎21 𝑎12 𝕧1 𝕧2 = 𝕠 𝕠 𝑓 − 𝑎22 𝑓 − 𝑎11 − 𝑎21 𝑎12 𝕧1 𝕧2 = 𝕠 𝕠 ∴ 𝑓2 − 𝑎11 + 𝑎22 𝑓 + 𝑎11 𝑎22 − 𝑎21 𝑎12 𝕧1 𝕧2 = 𝕠 𝕠

11. 𝑓2 − 𝑎11 + 𝑎22 𝑓 + 𝑎11 𝑎22 − 𝑎21 𝑎12 is Characteristic polynomial !

Characteristic polynomial

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Characteristic polynomial