Solve the following system of equations. x n + 1 = 2 y n − z n y n + 1 = y n z n + 1 = x n − 2 y n + 2 z n What is the solution in general for x 0 , y 0 , z 0 arbitrary?It is intended to be solved using Jordan Normal Form of the Linear Algebra knowledge. I have no idea how to start, can anyone give a hint?

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Solve the following system of equations.      x          n      +      1        =  2      y    n    −      z    n        y          n      +      1        =      y    n        z          n      +      1        =      x    n    −  2      y    n    +  2      z    n  What is the solution in general for       x    0  ,       y    0  ,       z    0   arbitrary?It is intended to be solved using Jordan Normal Form of the Linear Algebra knowledge. I have no idea how to start, can anyone give a hint?

hopeloothab9m · Accepted Answer

Hint. Note that for   n  ≥  0,      (                                        x                          n                                                                        y                          n                                                                        z                          n                                            )    =      (                            0                          2                          −          1                                      0                          1                          0                                      1                          −          2                          2                      )        (                                        x                          n              −              1                                                                        y                          n              −              1                                                                        z                          n              −              1                                            )    =            (                                    0                                2                                −            1                                                0                                1                                0                                                1                                −            2                                2                              )        n        (                                        x                          0                                                                        y                          0                                                                        z                          0                                            )    .Now find the Jordan normal form   J of the matrix  M  :=      (                            0                          2                          −          1                                      0                          1                          0                                      1                          −          2                          2                      )  and a matrix   P such that   M  =  P  J      P          −      1      . Then       M    n    =  P      J    n        P          −      1      .

Jamiya Weber · Accepted Answer

By the @Robert Z answer the jordan form of matrix   M is as follows:  M  =  P    J        P          −      1      where the matices   P and   J are in the following form;  P  =      (                            −          1                          3                          2                                      0                          1                          1                                      1                          0                          0                      )      ,    J  =      (                            1                          1                          0                                      0                          1                          0                                      0                          0                          1                      )  with the induction on   n you can prove that the   n-th power of matrix   J is as follows:      J    n    =      (                            1                          n                          0                                      0                          1                          0                                      0                          0                          1                      )  So, the   n-th power of matrix   M is in the following form:      M    n    =  P        J    n          P          −      1        =      (                            1          −          n                          2                    n                          −          n                                      0                          1                          0                                      n                          −          2                    n                          1          +          n                      )

Solve the following system of equations. x <mrow class="MJX-TeXAtom-ORD"> n

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