18.034, Honors Differential Equations Prof. Jason Starr Lecture 30: Notes for Kiran: I. Summary of Lecture 29 4/21/04 We finished last time with Jordan normal form. Notation(: ) a finite dimensional CI -vector space (usually = CI with standard basis ). inV V e ,....,e1 n (ii) T= → a linear operator V V (iii) A linear system of differential equations Tyy =' i.e. looking for differentiable y= IR → s.t. V () ()()tyTty ='. (iv) matrix of Tw.r.t. an ordered basis ()nvvB ,.....,1=, [],,BBTA= . iniijjvATv ⋅=∑=1(v) char. polynomial ()()()......det +−=−=λλλλTTTIdprnT v (vi) factorization () ( )()tmtmTpλλλλλ−−= .....11. (vii) for each ti ,....,1=, the generalized eigenspaces, )(= , riλ()riesIdTKλ−V = …… = iλ)1(iλ−C)2(iλ−C−C)(imiλgeniλ← generalized eigenspace. V V V V V (viii) restriction of Tto is . geniλiTV (ix) iN = - iT Idiλ . geniλV (x) Jordan normal form of : we saw that there is a sequence of lin. ind. sets of vectors (of course not every integer r has to occur above) iN() ( ) () ( ) () (1,11,1,1,,1,....,,....,,....,,...,,....,ararreaeeBBBBBB)raB,1such that = ) is an ordered basis for with respect to which we have iBλ()1,eBuu........( geiλnV []iBiNλ, = iBλ 18.034, Honors Differential Equations Page 1 of 9 Prof. Jason Starr NNNNNNNNN--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )1o--------------------(e,1)(e,1)(e,1)(e,1)(e,1)(r,j)(r,j)(r,j)(r,j)(r,j)(1,a )(1,a )(1,a )(1,a )(1,a )(1,a )(1,a
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