Suppose b is an n-dimensional column vector that can be decomposed as b = b 1 + b 2 . In this decomposition, b 1 and b 2 hold the non-zero components of b respectively (e.g. if b= [ 1 2 3 ] ⊤ , then this decomposition can be b 1 = [ 1 0 3 ] ⊤ and b 2 = [ 0 2 0 ] ⊤ ).Suppose A is a real matrix of order n and let C=[ b, A b , ... , A n − 1 b], C 1 =[ b 1 , A b 1 , ... , A n − 1 b 1 ] and C 2 =[ b 2 , A b 2 , ... , A n − 1 b 2 ]. If we have rank([ C 1 , C 2 ])=n, can we prove rank(C)=n?

Question

Suppose b is an n-dimensional column vector that can be decomposed as       b    =            b        1    +            b        2  . In this decomposition,             b        1   and             b        2   hold the non-zero components of   b respectively (e.g. if   b=             [                                    1                                2                                3                              ]        ⊤  , then this decomposition can be       b          1      =            [                                    1                                0                                3                              ]        ⊤   and   b      2  =            [                                    0                                2                                0                              ]        ⊤  ).Suppose A is a real matrix of order n and let   C=[  b,   A  b , ... ,       A          n      −      1        b],       C          1      =[      b          1      ,  A      b          1      , ... ,       A          n      −      1            b          1      ] and       C          2      =[      b          2      ,  A      b          2      , ... ,       A          n      −      1            b          2      ]. If we have rank([      C          1      ,      C          2      ])=n, can we prove rank(C)=n?

Xavier Jennings · Accepted Answer

The answer is no. As an example, consider  A  =      (                            1                          1                                      1                          1                      )    ,    b  =      (                            1                                      −          1                      )    ,        b    1    =      (                            1                                      0                      )    ,        b    2    =      (                            0                                      −          1                      )    .We find that       C    1    =  [      b    1          A      b    1    ] and       C    2    =  [      b    2          A      b    2    ] each have full row rank n=2, as does the block matrix   [      C    1              C    2    ]. However,   C  =  [  b        A  b  ] has a zero column and therefore does not have full rank.

Suppose b is an n-dimensional column vector that can be decomposed as b=b_1+b_2

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