Consider the system of linear differential equations: X = AX. (i) Suppose is an eigen value of A of multiplicity k > 1. Define a root vector corresponding to the eigen value \. (ii) Show that each eigen value \ of A of multiplicity k has k linearly independent root vectors. (iii) Suppose that A is an eigen value of A of multiplicity k and Uλ,1, Ux‚2, ……., Ux,k are root vectors corresponding to λ of orders 1, 2, k, respectively. Further, let: ...9 2! Xi(t) = e¼ [Ux,i + tŪλ,i−1 + U\‚i−2 + …... Show that X1(t), X2(t), . . ., Xk(t) {X₁(t), X2(t),...,x(t)} ti-1 (i − 1)! - ¡Uλ,1], for i = 1, 2, … . . k. is a set of linearly independent solutions of the system of linear differential equations and every solution of the system is a linear combination of the solutions in this set.

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