Given the linear transformation L : R3 ? R3 = (x1 + x2,x1 + x2 + x3,x2 + x3)^T ,

Question

Given the linear transformation L : R3 ? R3 = (x1 + x2,x1 + x2 + x3,x2 + x3)^T , find the matrix representation of A for unit vector basis. Suppose this linear transformation governs the equation Xn+1= AXn, and consider the new basis u1 = (1/2, -v2/2, 1/2)^T , u2 = (-v2/2, 0,v2/2)^T , u3 = (1/2,v2/2, 1/2)^T . What do you expect the long-time behavior of x be if the vector is aligned with u1, u2 and u3? Suppose that the initial state vector is x0 = (1, 1, 1)^T , where do you expect the state vector would align to as time n -> 8?

Explanation / Answer

see

http://www.math.tamu.edu/~stecher/LinearAlgebraPdfFiles/exercisesChap3.pdf

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