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Given the following matrix:

  mat = {{{2013, 9, 10, 0, 0, 0.}, "Tuesday", 10., "September", 2013., 10., 10., 1293., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 11, 0, 0, 0.}, "Wednesday", 11., "September", 2013., 11., 11., 1146., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 12, 0, 0, 0.}, "Thursday", 12., "September", 2013., 12., 12., 974., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 13, 0, 0, 0.}, "Friday", 13., "September", 2013., 13., 13., 1347., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 16, 0, 0, 0.}, "Monday", 16., "September", 2013., 16., 16., 1207., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 17, 0, 0, 0.}, "Tuesday", 17., "September", 2013., 17., 17., 1152., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 18, 0, 0, 0.}, "Wednesday", 18., "September", 2013., 18., 18., 1078., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 19, 0, 0, 0.}, "Thursday", 19., "September", 2013., 19., 19., 998., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 20, 0, 0, 0.}, "Friday", 20., "September", 2013., 20., 20., 1220., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 
         {{2013, 9, 23, 0, 0, 0.}, "Monday", 23., "September", 2013., 23., 23., 1096., 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}
        }

I also have the following matrix:

daymat = {{"Tuesday", 0., 0.0405246, 0.0475514, 0.055977, 0.0658123, 0.0620308, 0.0689194, 0.0653811, 0.074846, 0.0806566, 0.0680625, 0.065265, 0.0607537, 0.0657239, 0.0619091, 0.0518029, 0.0414976, 0.0232864, 0., 0., 0., 0., 0., 0., 0.}, 
          {"Wednesday", 0., 0.0389784, 0.0476981, 0.0553875, 0.0648306, 0.0645128, 0.0712814, 0.0666393, 0.0726954, 0.0765757, 0.0664365, 0.0629837, 0.0617724, 0.0656363, 0.0637674, 0.0558095, 0.0397895, 0.0252055, 0., 0., 0., 0., 0., 0., 0.}, 
          {"Thursday", 0., 0.0400184, 0.0478547, 0.0573118, 0.0680178, 0.0671116, 0.0719796, 0.0670419, 0.0745704, 0.0793687, 0.0660718, 0.0655258, 0.0612155, 0.0629117, 0.0615814, 0.048912, 0.0385196, 0.0219872, 0., 0., 0., 0., 0., 0., 0.}, 
          {"Friday", 0., 0.0393414, 0.047718, 0.0577941, 0.0642745, 0.060941, 0.0668365, 0.0616773, 0.0739372, 0.0744969, 0.0637601, 0.0628322, 0.0617429, 0.0604569, 0.0634777, 0.0573806, 0.049004, 0.0343285, 0., 0., 0., 0., 0., 0., 0.}, 
          {"Monday", 0., 0.0389818, 0.0466718, 0.0529945, 0.0614266, 0.0625077, 0.0673928, 0.064997, 0.0759944, 0.0822528, 0.0678311, 0.0643834, 0.0607663, 0.064325, 0.0655287, 0.0557701, 0.0422659, 0.02591, 0., 0., 0., 0., 0., 0., 0.}
         }

I want to join daymat to mat under the condition that day of week in daymat has to match with day of week in mat. For example, the 1st row of daymat, which is "Tuesday" will join with the 1st row in mat, which has the same day of week ("Tuesday").

How can I do that ?

Thanks

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Map[Function[arg, Append[arg, Select[daymat, #[[1]] == arg[[2]] &]]], mat] will "join" them, tinker as needed to get format you want.

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Replace[mat, {x_, z : #, y__} :> {x, z, y, ##2} & @@@ daymat, 1] 
{{{2013,9,10,0,0,0.},Tuesday,10.,September,2013.,10.,10.,1293.,0,0, <<39>>, 0.0414976, 
  0.0232864,0.,0.,0.,0.,0.,0.,0.},<<8>>,{<<1>>}}
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Is this what you mean?

Map[FlattenAt[Append[#, Cases[daymat, {#[[2]], n___} -> n, {1}, 1]], -1] &, mat]
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