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Unformatted text preview: Repeated Eigenvalues Recall We are now in the position that we can find (at least) as many linearly-independent 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 linearly-independent 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 linearly-independent 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 linearly-independent solutions as we have distinct eigenvalues. But it may be that the eigenvalues are repeated so that we can only find k linearly-independent solutions from eigenvectors in the manner described above. Since we need n linearly-independent 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 linearly-independent 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 ....
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