Transformari liniare
Transformari liniare
Transformari liniare
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No¸tiunea de transformare liniară<br />
Transformări <strong>liniare</strong> între spa¸tii finit dimensionale<br />
Valori ¸si vectori proprii<br />
Ecua¸tia caracteristică<br />
Se ob¸tine<br />
⎛<br />
a11 − λ<br />
⎜ a21 ⎜<br />
⎝ · · ·<br />
a12<br />
a22 − λ<br />
· · ·<br />
· · ·<br />
a1n<br />
a2n<br />
an1 an2 · · · ann − λ<br />
Diagonalizarea matricei unei transformări<br />
Polinom caracteristic<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
x1<br />
x2<br />
· · ·<br />
xn<br />
⎞<br />
⎟<br />
⎠ =<br />
⎛<br />
⎜<br />
⎝<br />
0<br />
0<br />
· · ·<br />
0<br />
⎞<br />
⎟<br />
⎠ .<br />
Sistemul are solu¸tie nebanală dacă<br />
<br />
a11 <br />
− λ<br />
a21 <br />
<br />
· · ·<br />
an1<br />
a12<br />
a22 − λ<br />
an2<br />
· · ·<br />
· · ·<br />
· · ·<br />
<br />
a1n <br />
<br />
a2n<br />
<br />
<br />
= 0<br />
<br />
ann − λ <br />
(8)<br />
Ecua¸tia (8) se nume¸ste ecua¸tie caracteristică.<br />
Transformări <strong>liniare</strong>
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