The multiplication of list of matrices

Question

I need to multiply 3 lists of matrices (b.a.b) as the following code

a = {{{-17.8227277373099`, -1.6565234964560602`}, 
{-1.6565234954649242`, 
 5.298073701591974`}}, {{-17.812203521929003`, 
-1.5013126607114478`}, {-1.5013126574896714`, 
 4.384050851253119`}}, {{-17.801677045750512`, 
 -1.4055541329078751`}, {-1.405554138172727`, 
 3.869511752542245`}}};

b = {{{0.8409518416651456`, 0}, {0, 
 0.1274293000222242`}}, {{0.8409815693580924`, 0}, {0, 
 0.14187218616724442`}}, {{0.841011296000238`, 0}, {0, 
 0.15290209433231844`}}};

I used the following:

  mat = b.a.b

But, I think this way is not correct because I didn't get the result. Also, I make a test for the first matrices as the following code, that what I wanted to get for the whole matrices multiplication.

 a1 = {{-17.8227277373099`, -1.6565234964560602`}, 
{-1.6565234954649242`, 5.298073701591974`}};

b1 = {{0.8409518416651456`, 0}, {0, 0.1274293000222242`}};

 mat1 = b1.a1.b1


 {{-12.6042, -0.177516}, {-0.177516, 0.0860313}}

Thanks.

Answer 1

Using MapThread and Dot:

MapThread[Dot, {b, a, b}]

Answer 2

If you need it really fast, then use Compile:

n = 1000000;
a = RandomReal[{-1, 1}, {n, 2, 2}];
b = RandomReal[{-1, 1}, {n, 2, 2}];

cf = Compile[{{a, _Real, 2}, {b, _Real, 2}},
   b.a.b,
   RuntimeAttributes -> Listable,
   Parallelization -> True
   ];

MapThread[Dot, {b, a, b}]; // AbsoluteTiming // First
cf[a, b]; // AbsoluteTiming // First

1.46343

0.075722