+-- | Returns True if the given matrix is upper-triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- False
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_upper_triangular m
+-- True
+--
+is_upper_triangular :: (Eq a, Ring.C a, Arity m, Arity n)
+ => Mat m n a -> Bool
+is_upper_triangular m =
+ and $ concat results
+ where
+ results = [[ test i j | i <- [0..(nrows m)-1]] | j <- [0..(ncols m)-1] ]
+
+ test :: Int -> Int -> Bool
+ test i j
+ | i <= j = True
+ | otherwise = m !!! (i,j) == 0
+
+
+-- | Returns True if the given matrix is lower-triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_lower_triangular m
+-- False
+--
+is_lower_triangular :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_lower_triangular = is_upper_triangular . transpose
+
+
+-- | Returns True if the given matrix is triangular, and False
+-- otherwise.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,0],[1,1]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[0,3]] :: Mat2 Int
+-- >>> is_triangular m
+-- True
+--
+-- >>> let m = fromList [[1,2],[3,4]] :: Mat2 Int
+-- >>> is_triangular m
+-- False
+--
+is_triangular :: (Eq a,
+ Ring.C a,
+ Arity m,
+ Arity n)
+ => Mat m n a
+ -> Bool
+is_triangular m = is_upper_triangular m || is_lower_triangular m
+
+
+-- | Return the (i,j)th minor of m.
+--
+-- Examples:
+--
+-- >>> let m = fromList [[1,2,3],[4,5,6],[7,8,9]] :: Mat3 Int
+-- >>> minor m 0 0 :: Mat2 Int
+-- ((5,6),(8,9))
+-- >>> minor m 1 1 :: Mat2 Int
+-- ((1,3),(7,9))
+--
+minor :: (m ~ S r,
+ n ~ S t,
+ Arity r,
+ Arity t)
+ => Mat m n a
+ -> Int
+ -> Int
+ -> Mat r t a
+minor (Mat rows) i j = m
+ where
+ rows' = delete rows i
+ m = Mat $ V.map ((flip delete) j) rows'
+
+
+class (Eq a, Ring.C a) => Determined p a where
+ determinant :: (p a) -> a
+
+instance (Eq a, Ring.C a) => Determined (Mat (S Z) (S Z)) a where
+ determinant m = m !!! (0,0)
+
+instance (Eq a, Ring.C a, Arity m) => Determined (Mat m m) a where
+ determinant _ = undefined
+
+instance (Eq a, Ring.C a, Arity n)
+ => Determined (Mat (S (S n)) (S (S n))) a where
+ determinant m
+ | is_triangular m = product [ m !!! (i,i) | i <- [0..(nrows m)-1] ]
+ | otherwise = determinant_recursive
+ where
+ m' i j = m !!! (i,j)
+
+ det_minor i j = determinant (minor m i j)
+
+ determinant_recursive =
+ sum [ (-1)^(1+(toInteger j)) NP.* (m' 0 j) NP.* (det_minor 0 j)
+ | j <- [0..(ncols m)-1] ]
+
+
+