The easiest normal systems to solve are systems of the form x\'(t) = Dx(t) where
ID: 3402665 • Letter: T
Question
The easiest normal systems to solve are systems of the form x'(t) = Dx(t) where D is an n Times n diagonal matrix. Such a system actually consists of n uncoupled equations x'_i (g) = d_ij x_i (t) I = 1...n. whose solution is x_i (t) = c_i e^where the c_i's are arbitrary constants. This raises the following question: When can we uncouple a normal system? To answer this question, we need the follow ing result from linear algebra. An n Times n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors P_1,...,P_n. Moreover, if P is the matrix whose columns are P_1...P_n, then P^-1 AP = D where D is the diagonal matrix whose entry d_ij is the eigenvalue associated with the vector p_i. Use the above result to show that the system x'(t) = Ax(t), where A is an n Times n diagonalizable matrix, is equivalent to an uncoupled system y'(t) = Dy(t), where y = P^-1 x and D = P^-1 AP.Explanation / Answer
Solution: Given that
x'(t) = Ax(t) (4)
where A is an nxn diagonalizable matrix.
Since A is diagonalizable matrix, so A has n linearly independent eigenvectors , say p1, p2,,.....,pn.
Take a nonsingular matrix P whose columns are the n linearly independent eigenvectors , p1, p2,,.....,pn.
Let y'(t) = Dy(t) (5)
where y = P-1x, D = P-1AP, D is the diagonal matrix whose principal diagonal entry is dii is the eigenvalue associated with the vector pi .
Since y'(t) = Dy(t) and y = P-1x,
we have P-1x'(t) = DP-1x(t)
implies that P(P-1x'(t)) = P(DP-1x(t)) (Pre multiplying by P)
implies that (PP-1)x'(t) = P(P-1AP)P-1x(t)( As D = P-1AP )
implies that (PP-1)x'(t) = (PP-1)A(PP-1)x(t)
implies that I .x'(t) = I.A.I x(t) (As PP-1 = I, identity matrix )
implies that x'(t) = A x(t)
Hence, by the given condition we can say that the system x'(t) = Ax(t) is equivalent to an uncoupled
system y'(t) = Dy(t) , where y = P-1x and D = P-1AP. (proved)
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