A Product function for matrix products

Question

I would like to use an equivalent of the product-function in mathematica, but where instead of multiplying numbers I multiply matrices?

Sometimes defining a new function for a simple task can make the code less readable. In this instance, I think that is the case because the syntax as shown by @DanielLichtblau is concise and sufficiently expressive to be self-explanatory. — Jens, Oct 27 '14 at 0:09

Dr. Wolfgang Hintze · Accepted Answer · 2014-10-25 19:59:19Z

Matrix multiplication is built in in Mathematica. Just use the dot for multiplication.

Here are two 2x2 matrices

a = PauliMatrix[1]
b = PauliMatrix[3]

(* Out[49]= {{0, 1}, {1, 0}} *)
(* Out[50]= {{1, 0}, {0, -1}} *)

Here's a product

a.b

(* Out[53]= {{0, -1}, {1, 0}} *)

and here is the product of the same factors in reverse order

b.a

(* Out[52]= {{0, 1}, {-1, 0}} *)

There are also two interesting functions of matrices

MatrixPower[a, 2]

(* Out[55]= {{1, 0}, {0, 1}} *)

MatrixExp[b]

(* Out[56]= {{E, 0}, {0, 1/E}} *)

Hope this helps,
Wolfgang

EDIT in response to the comment

I'm not aware of a generalization of Product[] to matrices in Mathematica. But you could easily define it yourself

matrixProduct[listOfMatrices_] := Dot @@ listOfMatrices

Then

m = Table[PauliMatrix[i], {i, 1, 3}];

matrixProduct[m]

(* Out[31]= {{I, 0}, {0, I}} *)

Regards,
Wolfgang

It is not exactly what I asked: I would like to make an iterated product but with matrices. E.g. Product[PauliMatrix[i],{i,1,3}], where the multiplication would be the dot . instead of the star *. — Anne O'Nyme, Oct 25 '14 at 19:03
Sorry, Anne. But see the EDIT in my answer. I hope it covers now what you want. — Dr. Wolfgang Hintze, Oct 25 '14 at 20:02

Mr.Wizard · Accepted Answer · 2014-10-26 05:25:43Z

Another method is to use Array, the fourth parameter of which sets the function that combines expressions:

m = RandomReal[9, {3, 3, 3}];

Array[m[[#]] &, 3, 1, Dot]

{{606.041, 638.877, 525.972},
 {1011.5, 1068.12, 856.671},
 {532.56, 556.236, 435.836}}

Equivalent to:

Dot @@ m

{{606.041, 638.877, 525.972},
 {1011.5, 1068.12, 856.671},
 {532.56, 556.236, 435.836}}

Mr.Wizard · Accepted Answer · 2014-10-26 07:31:09Z

productF = Block[{Times = Dot}, Product[#, #2]] &;

Examples:

m = Table[PauliMatrix[i], {i, 1, 3}];

productF[j, {j, m}]
(* {{I, 0},{0, I}} *)

productF[PauliMatrix[j], {j, 3}]
(* {{I, 0},{0, I}} *)

For better emulation of Product one may include the HoldAll attribute and accept additional Product iterators with:

productF2 = Function[, Block[{Times = Dot}, Product @ ##], HoldAll];

3 Answers 3