Can I simplify the following vector?Let a and b be n × 1 vectors and M 1 , M 2 , . . . , M n be n × n matrices.Is there a way to write the following vector: v = [ a T M 1 b a T M 2 b ⋮ a T M n b ] in such a way that reads v = a T M n e w b, where M n e w is just a manipulation of the M i matrices?I know I can do: v = [ a T M 1 a T M 2 ⋮ a T M n ] bBut then I cannot get the vector a "out of the brackets".

Question

Can I simplify the following vector?Let a and b be   n  ×  1 vectors and       M    1    ,      M    2    ,  .  .  .  ,      M    n    be   n  ×  n matrices.Is there a way to write the following vector:  v  =      [                                        a            T                                M            1                    b                                                  a            T                                M            2                    b                                      ⋮                                                  a            T                                M            n                    b                      ]  in such a way that reads   v  =      a    T        M          n      e      w        b, where       M          n      e      w        is just a manipulation of the       M    i   matrices?I know I can do:  v  =      [                                        a            T                                M            1                                                            a            T                                M            2                                                ⋮                                                  a            T                                M            n                                ]    bBut then I cannot get the vector a &quot;out of the brackets&quot;.

sviudes7w · Accepted Answer

You could write  v  =      (                                        a            T                                                        ⋱                                                                          a            T                                )        (                                        M            1                                                ⋮                                                  M            n                                )    b  .If you like, you could write the matrix on the left compactly using the fact that      (                                        a            T                                                        ⋱                                                                          a            T                                )    =      a    T    ⊗  I  ,where I denotes a size n identity matrix and ⊗ denotes the Kronecker product.