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  1. PO
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    <p>It'll be perfectly safe with a smart constructor and stored dimensions. Of course there are no natural implementations for the operations <code>signum</code> and <code>fromIntegral</code> (or maybe a diagonal matrix would be fine for the latter).</p> <pre class="lang-hs prettyprint-override"><code>module Matrix (Matrix(),matrix,matrixTranspose) where import Data.List (transpose) data Matrix a = Matrix {matrixN :: Int, matrixM :: Int, matrixElems :: [[a]]} deriving (Show, Eq) matrix :: Int -&gt; Int -&gt; [[a]] -&gt; Matrix a matrix n m vals | length vals /= m = error "Wrong number of rows" | any (/=n) $ map length vals = error "Column length mismatch" | otherwise = Matrix n m vals matrixTranspose (Matrix m n vals) = matrix n m (transpose vals) instance Num a =&gt; Num (Matrix a) where (+) (Matrix m n vals) (Matrix m' n' vals') | m/=m' = error "Row number mismatch" | n/=n' = error "Column number mismatch" | otherwise = Matrix m n (zipWith (zipWith (+)) vals vals') abs (Matrix m n vals) = Matrix m n (map (map abs) vals) negate (Matrix m n vals) = Matrix m n (map (map negate) vals) (*) (Matrix m n vals) (Matrix n' p vals') | n/=n' = error "Matrix dimension mismatch in multiplication" | otherwise = let tvals' = transpose vals' dot x y = sum $ zipWith (*) x y result = map (\col -&gt; map (dot col) tvals') vals in Matrix m p result </code></pre> <p>Test it in ghci:</p> <pre class="lang-hs prettyprint-override"><code>*Matrix&gt; let a = matrix 3 2 [[1,0,2],[-1,3,1]] *Matrix&gt; let b = matrix 2 3 [[3,1],[2,1],[1,0]] *Matrix&gt; a*b Matrix {matrixN = 3, matrixM = 3, matrixElems = [[5,1],[4,2]]} </code></pre> <p>Since my <code>Num</code> instance is generic, it even works for complex matrices out of the box:</p> <pre class="lang-hs prettyprint-override"><code>Prelude Data.Complex Matrix&gt; let c = matrix 2 2 [[0:+1,1:+0],[5:+2,4:+3]] Prelude Data.Complex Matrix&gt; let a = matrix 2 2 [[0:+1,1:+0],[5:+2,4:+3]] Prelude Data.Complex Matrix&gt; let b = matrix 2 3 [[3:+0,1],[2,1],[1,0]] Prelude Data.Complex Matrix&gt; a Matrix {matrixN = 2, matrixM = 2, matrixElems = [[0.0 :+ 1.0,1.0 :+ 0.0],[5.0 :+ 2.0,4.0 :+ 3.0]]} Prelude Data.Complex Matrix&gt; b Matrix {matrixN = 2, matrixM = 3, matrixElems = [[3.0 :+ 0.0,1.0 :+ 0.0],[2.0 :+ 0.0,1.0 :+ 0.0],[1.0 :+ 0.0,0.0 :+ 0.0]]} Prelude Data.Complex Matrix&gt; a*b Matrix {matrixN = 2, matrixM = 3, matrixElems = [[2.0 :+ 3.0,1.0 :+ 1.0],[23.0 :+ 12.0,9.0 :+ 5.0]]} </code></pre> <hr> <p><strong>EDIT: new material</strong></p> <p>Oh, you want to just override the <code>(*)</code> function without any <code>Num</code> stuff. That's possible to o but you'll have to remember that the Haskell standard library has reserved <code>(*)</code> for use in the <code>Num</code> class.</p> <pre class="lang-hs prettyprint-override"><code>module Matrix where import qualified Prelude as P import Prelude hiding ((*)) import Data.List (transpose) class Multiply a where (*) :: a -&gt; a -&gt; a data Matrix a = Matrix {matrixN :: Int, matrixM :: Int, matrixElems :: [[a]]} deriving (Show, Eq) matrix :: Int -&gt; Int -&gt; [[a]] -&gt; Matrix a matrix n m vals | length vals /= m = error "Wrong number of rows" | any (/=n) $ map length vals = error "Column length mismatch" | otherwise = Matrix n m vals matrixTranspose (Matrix m n vals) = matrix n m (transpose vals) instance P.Num a =&gt; Multiply (Matrix a) where (*) (Matrix m n vals) (Matrix n' p vals') | n/=n' = error "Matrix dimension mismatch in multiplication" | otherwise = let tvals' = transpose vals' dot x y = sum $ zipWith (P.*) x y result = map (\col -&gt; map (dot col) tvals') vals in Matrix m p result a = matrix 3 2 [[1,2,3],[4,5,6]] b = a * matrixTranspose </code></pre> <p>Testing in ghci:</p> <pre class="lang-hs prettyprint-override"><code>*Matrix&gt; b Matrix {matrixN = 3, matrixM = 3, matrixElems = [[14,32],[32,77]]} </code></pre> <p>There. Now if a third module wants to use both the <code>Matrix</code> version of <code>(*)</code> and the <code>Prelude</code> version of <code>(*)</code> it'll have to of course import one or the other qualified. But that's just business as usual.</p> <p>I could've done all of this without the <code>Multiply</code> type class but this implementation leaves our new shiny <code>(*)</code> open for extension in other modules.</p>
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    1. CO__"Haskell standard library has reserved (*) for use in the Num class"__ That is EXACTLY what I wanted to know. I wanted to know how "flexible" Haskell was before I started learning it. Thank you. Wow, this took really long time to figure out.
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    2. CO@drozzy: also, I'll add that Haskellers tend to use new operators (like the `|+|` from your previous question) instead of overloading when overloading isn't exactly the right thing. This is a big contrast to languages like C++ or Python where the set of operators is limited so people have to keep reusing (or actually misusing) them.
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    3. COI think Haskellers use new operators way too often. But besides that it seems to me that they "defend" their use of new operators as being right - when the fact is they have no choice. So they answer to my question (this and previous) should have been: NO, you cannot override + (or *). Period.
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