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Linear systems of ordinary differential equations (ODEs) with matrix-valued functions. It covers the concepts of eigenvalues and eigenvectors, their significance, and how to find them. The document also discusses general solutions of linear systems and the Wronskian determinant.
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Linear (^) system of ODES An Inhoduchbn a (^) Matrix_valued functions and systems (^) of odes Eigenvalues and Eigenvectors a (^) General solutions (^) of linear (^) systems
systemsof__0DI_iLetAltlbeamatrix valued function
a ⼼ (^92) ⼼ (^) anltl Gultl (^) Gultl 92314 siltl aznltl sltl he define^ Ait to^ be^ the^ matrix^ whose^ ijtth element is 璺 劉 ij ⼆ 㼻 Similarly we^ define^ f Alt^ dt^ to be the^ matrix whose Cij th^ element^ is faijctldt.PAltldt faijltldt.li
0 Def linear^ ODE A (^) system of (^) system of (^) H orderODE.is h巡 if it can be written^ as X ⼆^ pyxtgdnmn vector maĚx column (^) vector For (^) example x (^) sinltlxttytcosltl y (^) cosctìytet
yfsintytf 0 Cosltl PHYsinltltzoo.nl
笑 7
x (^) A b
EigenraluesandEigenvectovs_re.co Why would^ we like (^) to (^) study eigenvalues eigenvectors Consider the (^) system XEAXXlt etv .at Then X^ t (^) rertr Xlt is^ a^ solution^ to^ x^ Ax rertzi Alert r ertnj Alrkrv
Let A^ be^ an nxn^ matrix A real (^) number 2 is an gender
exists a^ nonzero 0such that Alv IT v is called^ an eigeng
o (^) How (^) do u find values Example
Yf We (^) want to^ find 九^ and nonzero^ v^ such^ that A v (^) to This is^ equivalent to CAXI r^0 I (^) is the (^) identity 2x2matrix and v^ is^ a^ nonzero^ vector Reall that Bx ⼆^0 has a nonzero so 7 det B 0 Therefore (^) 入 is (^) an eigenvalue if andonly^ if det A 入到 ⼆ 0 det (^) 印 品 0 ⼈才 2 (^4 0) n 三^4 1 ⼆ (^) 上 (^2) Hence ⼑ ⼆^ ⼗ (^023)
⻔
Eigenvectors (^) of 九 ⼆ (^3) i Solve A 2 ⼯^ v 0 (^1 3 ) 1 1 31 台 (^1) 州 i 倒^ 州 V1 2Vz ⼆^0 不比^ V2^ ⼆^1 then^ V1^ ⼆^2 以^ ⼆ (^2) i is an (^) eigenvector 在 0 與
non zero^ number If (^) vto is (^) an eigenvector of 入 then (^) so is CU When you find eigenvectors of a zxz^ matrix the two^ eqs from^ AND no shouldbe equivalent
X Ax has (^) at (^) least two solutions ētfilandl Q Can^ you produce^ more^ solutions^ from^ the two solutions^ above yes Get (^) Hl is^ also^ a solution.ae 引 is also a^ solution so is^ a^ etlytaeyy
(^0) Wronskian Def Let 如^ ⼼ 中⼼^ XMH he^ solutions (^) to the homogeneous eq X^ Plt x The Wronskian^ of (^) X It MH is defined^ as WEx^ 如 了 (^) ⼀ det 炸^ X刻^ ⼀ (^) Mtl i This is (^) the matrix whose columns (^) are 炒炒 X Example Consider the (^) equation x Y X From (^) the (^) process above 灿 et^ H X t^ eikl me solutions 2e3thnx 如火 detf 2et et et
⼆ 2ezt_^2 ezt ⼆ (^4) e2t (^) to a (^) Theorem (^) General solutions let PH be an nxn matrix_valued^ function If 如^ t^ X呢^ Mctl^ me^ solutions^ to
with (^) nonzero Wronskian^ for (^) all t (^) then the general solution^ of^ Cts^ is
呢 (^) GÜHI By the theorem^ above the (^) general solution of x (^) Y x^ is aēt Htsei 引
iifòǒǒon iǒ 1 lòiiii free variable V3 ⼆^ t 3t.li 6t 引 is an eigenvector 九⼆ soul (^) ill 1 (^1 ) 0i iii liiiii fiiǒig 0 0 0lots free variable it (^) n_n 何 is an eigenvector
入 i (^) solve 2 4 o ⼀
1 1 1 (^2 ) 0i 0 (^1 1 0 ) 1 1 1 0
20i 8
o 3 1 0
o 1 (^0 3 11 ) Rze B o o^ o^ i O ㄩ (^) free V32t h^ I h^ 㕺 take (^) ts
is an^ eigenvector
