This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Repeated Eigenvalues Recall We are now in the position that we can find (at least) as many linearlyindependent solutions to the homogeneous equation y = A y as there are distinct eigenvalues of A (real or complex). This is because for each real eigenvalue we can try to find as many linearlyindependent eigenvectors v as we can by solving ( A I ) v = 0 and for each one we find we have a solution of the form y = e x v . Similarly, for each complex eigenvalue = + i we find, we can find an eigenvector v = v real + i v im by solving ( A I ) v = 0 and this produces two linearlyindependent solutions y 1 = e x cos( x ) v real sin( x ) v im and y 2 = e x sin( x ) v real + cos( x ) v im . Thus we have (at least) as many linearlyindependent solutions as we have distinct eigenvalues. But it may be that the eigenvalues are repeated so that we can only find k linearlyindependent solutions from eigenvectors in the manner described above. Since we need n linearlyindependent solutions y 1 , y 2 ,..., y n to write down the general solution y ( x ) = c 1 y ! + c 2 y 2 + + c n y n we need to find another n k linearlyindependent solutions. Generalized Eigenvectors Before we discuss a way of producing more solutions, we need a few definitions. Definition The matrix exponential function e X : M n ( R ) M n ( R ) is a function that takes an n n matrix X as an input and outputs a new n n matrix denoted e X . The matrix e X is defined by substituting X into the Taylor series for e x , which gives e X = I + X + X 2 2! + X 3 3! + where I is the identity matrix. The series e X converges for all n n matrices X ....
View Full
Document
 '08
 staff

Click to edit the document details