# Multiply list of matrices by list of vectors

I have a list of vectors and a list of matrices for each vector, what i need is to multiply each list of matrices to respective vector.

v = {{a1, a2}, {b1, b2}, {c1, c2}};

U = {
{{{U1a11, U1a12}, {U1a21, U1a22}},
{{U2a11, U2a12}, {U2a21, U2a22}},
{{U3a11, U3a12}, {U3a21, U3a22}},
{{U4a11, U4a12}, {U4a21, U4a22}}}},
{{{U1b11, U1b12}, {U1b21, U1b22}},
{{U2b11, U2b12}, {U2b21, U2b22}},
{{U3b11, U3b12}, {U3b21, U3b22}},
{{U4b11, U4b12}, {U4b21, U4b22}}},
{{{U1c11, U1c12}, {U1c21, U1c22}},
{{U2c11, U2c12}, {U2c21, U2c22}},
{{U3c11, U3c12}, {U3c21, U3c22}},
{{U4c11, U4c12}, {U4c21, U4c22}}}
};


I know that I can use a Do

vt = ConstantArray[0, {Length[U], Length[v]}];

Do[vt[[it]] = (#.v[[it]]) & /@ U[[it]], {it, 1, 3}]

vt


But I don't want to use a Do cycle. Is there a way to do it in an intelligent way?

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A few alternatives to MapThread, with res = MapThread[Dot, {U, v}] from Dr. belisarius' answer,

res1 = Dot @@@ Transpose[{U, v}];
res2 = Dot @@@ Thread[{U, v}];
res3 = Thread[dot[U, v]] /. dot -> Dot;
res4 = Activate[Thread[Inactive[Dot][U,v]]] (* if you have version 10 *);
Equal @@ {res1, res2, res3, res4, res}


True

• +1 I particularly hate the Thread[dot[U, v]] /. dot -> Dot thing. It is needed in a lot of situations (like the testxxxQ functions) and it makes me feel soooo uncomfortable Mar 21, 2016 at 10:52
• @Dr.belisarius I can't agree more. Thank you for the vote.
– kglr
Mar 21, 2016 at 11:11
v = {{a1, a2}, {b1, b2}, {c1, c2}};

U = {
{{{U1a11, U1a12}, {U1a21, U1a22}}, {{U2a11, U2a12}, {U2a21, U2a22}},
{{U3a11, U3a12}, {U3a21, U3a22}}, {{U4a11, U4a12}, {U4a21, U4a22}}},
{{{U1b11, U1b12}, {U1b21, U1b22}}, {{U2b11, U2b12}, {U2b21, U2b22}},
{{U3b11, U3b12}, {U3b21, U3b22}}, {{U4b11, U4b12}, {U4b21,U4b22}}},
{{{U1c11, U1c12}, {U1c21, U1c22}}, {{U2c11, U2c12}, {U2c21, U2c22}},
{{U3c11, U3c12}, {U3c21, U3c22}}, {{U4c11, U4c12}, {U4c21, U4c22}}}};

res = MapThread[#1.#2 &, {U, v}];

Map[MatrixForm, res, {2}]


• Why not just Dot instead of wrapping it in an additional pure function call with #1.#2&? Mar 21, 2016 at 11:16
• @MartinBüttner No special reason. I feel more comfortable with it, and the problem is small enough to hardly justify trying to measure performance subtleties Mar 21, 2016 at 11:41
• @MartinBüttner Now, thinking again about why I tend to use it like this, see kglr's answer with the Thread[dot[U, v]] /. dot -> Dot horrible construction. Thread and Dot are a nasty marriage and I guess that one develops allergies to the simple sight of them together. I'm using MapThread here, but probably my running nose doesn't discriminate Mar 21, 2016 at 11:46