You'll need to define lambda first, otherwise it's trying to take elements from a matrix that doesn't exist. You'll also need to have your other parameters defined if you want a numeric output instead of a symbolic expression. As an example, if I had the following (randomly generated) values for lambda
\[Lambda] = RandomReal[1, {4, 5}] (*using M=5 here*)
{{0.677488,0.632777,0.248209,0.0187061,0.193736},
{0.115021,0.632859,0.931393,0.730297,0.298366},
{0.204479,0.266129,0.831798,0.689316,0.55112},
{0.301503,0.965564,0.208395,0.0999985,0.495401}}
The product gives
If there were also values defined for s, n, r, and k1, it would evaluate to a single real number.
To visualize how it works, I can set up symbolic values for lambda like so:
\[Lambda] = Table[
Symbol[ToString[\[Lambda]] <> ToString[k] <> ToString[m]],
{k, 1, 4}, {m, 1, 5}] (*again using M=5, but it could be any natural number*)
This creates a symbolic matrix with the following structure:
{{\[Lambda]11, \[Lambda]12, \[Lambda]13, \[Lambda]14, \[Lambda]15},
{\[Lambda]21, \[Lambda]22, \[Lambda]23, \[Lambda]24, \[Lambda]25},
{\[Lambda]31, \[Lambda]32, \[Lambda]33, \[Lambda]34, \[Lambda]35},
{\[Lambda]41, \[Lambda]42, \[Lambda]43, \[Lambda]44, \[Lambda]45}}
Also, since M is dependent on the number of columns of lambda, it makes sense to just define it as
M = Length[Transpose[\[Lambda]]]
Then evaluation of the product
Product[
1 - 4 (2^(2 r) - 1)/(s n r) Sqrt[
Product[\[Lambda][[k, m]], {k, 1, 4}]
] k1 (Sqrt[
4 \[Lambda][[1, m]] \[Lambda][[2, m]] (2^(2 r) - 1)/(s n r)
]) k1 (Sqrt[
4 \[Lambda][[3, m]] \[Lambda][[4, m]] (2^(2 r) - 1)/(s n r)
]), {m, 1, M}]
then gives the following output:
(1 - (16 (-1 + 2^(
2 r)) k1^2 Sqrt[((-1 + 2^(2 r)) \[Lambda]11 \[Lambda]21)/(
n r s)] Sqrt[((-1 + 2^(2 r)) \[Lambda]31 \[Lambda]41)/(n r s)]
Sqrt[\[Lambda]11 \[Lambda]21 \[Lambda]31 \[Lambda]41])/(
n r s)) (1 - (
16 (-1 + 2^(
2 r)) k1^2 Sqrt[((-1 + 2^(2 r)) \[Lambda]12 \[Lambda]22)/(
n r s)] Sqrt[((-1 + 2^(2 r)) \[Lambda]32 \[Lambda]42)/(n r s)]
Sqrt[\[Lambda]12 \[Lambda]22 \[Lambda]32 \[Lambda]42])/(
n r s)) (1 - (
16 (-1 + 2^(
2 r)) k1^2 Sqrt[((-1 + 2^(2 r)) \[Lambda]13 \[Lambda]23)/(
n r s)] Sqrt[((-1 + 2^(2 r)) \[Lambda]33 \[Lambda]43)/(n r s)]
Sqrt[\[Lambda]13 \[Lambda]23 \[Lambda]33 \[Lambda]43])/(
n r s)) (1 - (
16 (-1 + 2^(
2 r)) k1^2 Sqrt[((-1 + 2^(2 r)) \[Lambda]14 \[Lambda]24)/(
n r s)] Sqrt[((-1 + 2^(2 r)) \[Lambda]34 \[Lambda]44)/(n r s)]
Sqrt[\[Lambda]14 \[Lambda]24 \[Lambda]34 \[Lambda]44])/(
n r s)) (1 - (
16 (-1 + 2^(
2 r)) k1^2 Sqrt[((-1 + 2^(2 r)) \[Lambda]15 \[Lambda]25)/(
n r s)] Sqrt[((-1 + 2^(2 r)) \[Lambda]35 \[Lambda]45)/(n r s)]
Sqrt[\[Lambda]15 \[Lambda]25 \[Lambda]35 \[Lambda]45])/(n r s))
(which looks nicer in Mathematica than it does on here).
