For matrices P ∈ R m × n and Q ∈ R n × d , let p i be the i t h column of P, and q i T the i t h line of Q. We want to show that P Q = ∑ i = 1 n p i q i T I saw a proof using the fact that ∑ k p k q k T = ∑ k ∑ i ∑ j p i k q k j M i j with M i j a matrix full of zeros everywhere except ( M i j ) i j = 1Is there a proof that does not use triple sums ?

Question

For matrices   P  ∈            R              m      ×      n       and   Q  ∈            R              n      ×      d      , let       p    i   be the       i          t      h       column of P, and       q    i    T   the       i          t      h       line of Q. We want to show that  P  Q  =      ∑          i      =      1        n        p    i        q    i    T  I saw a proof using the fact that       ∑    k        p    k        q    k    T    =      ∑    k        ∑    i        ∑    j        p          i      k            q          k      j            M          i      j       with       M          i      j       a matrix full of zeros everywhere except   (      M          i      j            )          i      j        =  1Is there a proof that does not use triple sums ?

n8ar1val · Accepted Answer

Let       ε    k   denote the canonical vector basis for                     R                    n      , then the indexed vectors can be written in terms of the underlying matrices as                              p          k                =        P                  ε          k                                          q          k          T                =                  e          k          T                Q            Therefore the sum in question can be expanded as                              ∑                      k            =            1                    n                          p          k                          q          k          T                            =                  ∑                      k            =            1                    n                P                  ε          k                          ε          k          T                Q                                  =        P                  (                      ∑                          k              =              1                        n                                ε            k                                ε            k            T                    )                Q                                  =        P                          I          n                        Q                                  =        P        Q

Danika Mckay · Accepted Answer

Check that the i,j entry of the left-hand side is the i,j entry of the right-hand side.If you think about how the matrix multiplication PQ works, you will note that the i,j entry of PQ is       ∑          k      =      1        n        p          i      k            q          k      j      For the right-hand side       ∑          k      =      1        n        p    k        q    k    ⊤  , note that       p          i      k            q          k      j       is the i,j entry of       p    k        q    k    ⊤

Show that PQ=sum_(i=1)^n p_i q_i^T

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