Outline - Purdue Universitystindel/teaching/ma262/...Generalized eigenvectors with multiplicity 2...
20
Outline 1 First order systems and applications 2 Matrices and linear systems 3 The eigenvalue method for linear systems Distinct eigenvalues Complex eigenvalues 4 Multiple eigenvalue solutions 5 A gallery of solution curves of linear systems Real eigenvalues Complex eigenvalues Samy T. Systems Differential equations 45 / 92
Outline - Purdue Universitystindel/teaching/ma262/...Generalized eigenvectors with multiplicity 2 System: xÕ = Ax,withA œ R2,2 and det(A) ”= 0 Situation: A has a double eigenvalue
3 The eigenvalue method for linear systemsDistinct eigenvaluesComplex eigenvalues
4 Multiple eigenvalue solutions
5 A gallery of solution curves of linear systemsReal eigenvaluesComplex eigenvalues
Samy T. Systems Di�erential equations 45 / 92
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Example of matrix with repeated rootMatrix:
A =A
1 ≠11 3
B
Characteristic polynomial:
PA(r) = det(A ≠ r Id) = (r ≠ 2)2
Eigenvalues and eigenvectors:
r = 2, v1 =A
1≠1
B
Remark: r = 2 is a double eigenvalue, with 1 eigenvector only.
Samy T. Systems Di�erential equations 46 / 92
Generalized eigenvectors with multiplicity 2
System: xÕ = Ax, with A œ R2,2 and det(A) ”= 0
Situation:A has a double eigenvalue rUnique eigenvector v (up to constant factor)
Recipe to find generalized eigenvectors:1 Find v2 such that (A ≠ r Id)2v2 = 0, but not parallel to v2 Compute v1 = (A ≠ r Id)v23 Then v1, v2 are generalized eigenvectors
Samy T. Systems Di�erential equations 47 / 92
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Solving systems with multiplicity 2
Situation:We consider the system xÕ = A xA has a double eigenvalue rGeneralized eigenvectors v1, v2
Corresponding fundamental solutions: We get
x1(t) = v1ert
x2(t) = (v1t + v2) ert
Samy T. Systems Di�erential equations 48 / 92
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Example with multiplicity 2 (1)
Equation:
xÕ =A
1 ≠11 3
B
x
Eigenvalues and eigenvector:
r = 2 (multiplicity 2), v =A
1≠1
B
Square of a matrix: We have
A ≠ 2 Id =C≠1 ≠11 1
D
, (A ≠ 2Id)2 = 0
Samy T. Systems Di�erential equations 49 / 92
Example with multiplicity 2 (2)
Applying the recipe to find the generalized eigenvectors: We choose
v2 =C10
D
, v1 = (A ≠ 2Id) v2 =C≠11
D
General solution:
x = c1
A≠1
1
B
e2t + c2
A≠1
1
B
te2t + c2
A10
B
e2t
Samy T. Systems Di�erential equations 50 / 92
Example with multiplicity 2 (3)Asymptotic behavior: As t æ Œ
x(t) æ ŒlimtæŒ
x2(t)x1(t) = ≠1, thus slope ƒ ≠1
x(t) does not approach the asymptote
Graph in the x1x2 plane:
Samy T. Systems Di�erential equations 51 / 92
Generalized eigenvectors with multiplicity 3
Situation:A has a triple eigenvalue rUnique eigenvector v (up to constant factor)
Recipe to find generalized eigenvectors:1 Find v3 such that (A ≠ r Id)3v3 = 0, not parallel to v2 Compute v2 = (A ≠ r Id)v33 Compute v1 = (A ≠ r Id)v24 Then v1, v2, v3 are generalized eigenvectors
Samy T. Systems Di�erential equations 52 / 92
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Mobile User
Solving systems with multiplicity 3
Situation:We consider the system xÕ = A xA has a triple eigenvalue rGeneralized eigenvectors v1, v2, v3