MATH 33B Lecture 19: Lecture 19

• School: UCLA
• Department: Mathematics
• Course: MATH 33B
• Professor: Visan

Document Summary

Consider (cid:1876) =(cid:1827)(cid:1876) where (cid:1876): , (cid:1827) is an matrix with constant real coefficients. We saw that (cid:1876)(cid:4666)(cid:4667)=(cid:1857)(cid:1874) is a solution if is an eigenvalue of (cid:1827) and (cid:1874) is an eigenvector with eigenvalue . In order for to be an eigenvalue of (cid:1827), there must exist a nonzero (cid:1874) such that (cid:1827)(cid:1874)=(cid:1874). (cid:4666)(cid:1856) (cid:1827)(cid:4667)(cid:1874)=(cid:882) So the question is: when does there exist a nonzero (cid:1874) such that (cid:1828)(cid:1874)=(cid:882)? (cid:1855) (cid:1856)(cid:4673),(cid:1874)=(cid:4672)(cid:1874)(cid:2869)(cid:1874)(cid:2870)(cid:4673) Think of (cid:4666)(cid:1845)(cid:4667) as a system of equations with 2 unknowns (cid:1874)(cid:2869),(cid:1874)(cid:2870) If det(cid:1828) (cid:882), there is a unique solution to (cid:4666)(cid:1845)(cid:4667) (cid:1874)(cid:2870)=det(cid:4672)(cid:1853) (cid:882)(cid:1855) (cid:882)(cid:4673) (cid:1874)(cid:2869)=det(cid:4672)(cid:882) (cid:1854)(cid:882) (cid:1856)(cid:4673) (cid:1855) (cid:1856)(cid:4673)=(cid:882), (cid:1855) (cid:1856)(cid:4673)=(cid:882) det(cid:4672)(cid:1853) (cid:1854) det(cid:4672)(cid:1853) (cid:1854) So, in order to have a (cid:1874) (cid:882) such that (cid:1828)(cid:1874)=(cid:882), we must have det(cid:1828)=(cid:882). Thus in order for to be an eigenvalue of (cid:1827), we must have det(cid:4666)(cid:1856) (cid:1827)(cid:4667)=(cid:882). The equation det(cid:4666)(cid:1856) (cid:1827)(cid:4667)=(cid:882) is called the characteristic equation.