cgmath/src/matrix.rs

// Copyright 2013-2014 The CGMath Developers. For a full listing of the authors,
// refer to the AUTHORS file at the top-level directory of this distribution.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//     http://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

//! Column major, square matrix types and traits.

use std::fmt;
use std::mem;
use std::num::{Zero, zero, One, one, cast};

use angle::{Rad, sin, cos, sin_cos};
use approx::ApproxEq;
use array::{Array1, Array2};
use num::{BaseFloat, BaseNum};
use point::{Point, Point3};
use quaternion::{Quaternion, ToQuaternion};
use vector::{Vector, EuclideanVector};
use vector::{Vector2, Vector3, Vector4};

/// A 2 x 2, column major matrix
#[deriving(Clone, PartialEq)]
pub struct Matrix2<S> { pub x: Vector2<S>, pub y: Vector2<S> }

/// A 3 x 3, column major matrix
#[deriving(Clone, PartialEq)]
pub struct Matrix3<S> { pub x: Vector3<S>, pub y: Vector3<S>, pub z: Vector3<S> }

/// A 4 x 4, column major matrix
#[deriving(Clone, PartialEq)]
pub struct Matrix4<S> { pub x: Vector4<S>, pub y: Vector4<S>, pub z: Vector4<S>, pub w: Vector4<S> }


impl<S: BaseNum> Matrix2<S> {
    /// Create a new matrix, providing values for each index.
    #[inline]
    pub fn new(c0r0: S, c0r1: S,
               c1r0: S, c1r1: S) -> Matrix2<S> {
        Matrix2::from_cols(Vector2::new(c0r0, c0r1),
                           Vector2::new(c1r0, c1r1))
    }

    /// Create a new matrix, providing columns.
    #[inline]
    pub fn from_cols(c0: Vector2<S>, c1: Vector2<S>) -> Matrix2<S> {
        Matrix2 { x: c0, y: c1 }
    }

    /// Create a new diagonal matrix, providing a single value to use for each
    /// non-zero index.
    #[inline]
    pub fn from_value(value: S) -> Matrix2<S> {
        Matrix2::new(value.clone(), zero(),
                     zero(), value.clone())
    }

    /// Create a zero matrix (all zeros).
    #[inline]
    pub fn zero() -> Matrix2<S> {
        Matrix2::from_value(zero())
    }

    /// Create an identity matrix (diagonal matrix of ones).
    #[inline]
    pub fn identity() -> Matrix2<S> {
        Matrix2::from_value(one())
    }
}

impl<S: BaseFloat> Matrix2<S> {
    /// Create a transformation matrix that will cause a vector to point at
    /// `dir`, using `up` for orientation.
    pub fn look_at(dir: &Vector2<S>, up: &Vector2<S>) -> Matrix2<S> {
        //TODO: verify look_at 2D
        Matrix2::from_cols(up.clone(), dir.clone()).transpose()
    }

    #[inline]
    pub fn from_angle(theta: Rad<S>) -> Matrix2<S> {
        let cos_theta = cos(theta.clone());
        let sin_theta = sin(theta.clone());

        Matrix2::new(cos_theta.clone(), -sin_theta.clone(),
                     sin_theta.clone(),  cos_theta.clone())
    }
}

impl<S: BaseNum> Matrix3<S> {
    /// Create a new matrix, providing values for each index.
    #[inline]
    pub fn new(c0r0:S, c0r1:S, c0r2:S,
               c1r0:S, c1r1:S, c1r2:S,
               c2r0:S, c2r1:S, c2r2:S) -> Matrix3<S> {
        Matrix3::from_cols(Vector3::new(c0r0, c0r1, c0r2),
                           Vector3::new(c1r0, c1r1, c1r2),
                           Vector3::new(c2r0, c2r1, c2r2))
    }

    /// Create a new matrix, providing columns.
    #[inline]
    pub fn from_cols(c0: Vector3<S>, c1: Vector3<S>, c2: Vector3<S>) -> Matrix3<S> {
        Matrix3 { x: c0, y: c1, z: c2 }
    }

    /// Create a new diagonal matrix, providing a single value to use for each
    /// non-zero index.
    #[inline]
    pub fn from_value(value: S) -> Matrix3<S> {
        Matrix3::new(value.clone(), zero(), zero(),
                     zero(), value.clone(), zero(),
                     zero(), zero(), value.clone())
    }

    /// Create a zero matrix (all zeros).
    #[inline]
    pub fn zero() -> Matrix3<S> {
        Matrix3::from_value(zero())
    }

    /// Create an identity matrix (diagonal matrix of ones).
    #[inline]
    pub fn identity() -> Matrix3<S> {
        Matrix3::from_value(one())
    }
}

impl<S: BaseFloat>
Matrix3<S> {
    /// Create a transformation matrix that will cause a vector to point at
    /// `dir`, using `up` for orientation.
    pub fn look_at(dir: &Vector3<S>, up: &Vector3<S>) -> Matrix3<S> {
        let dir = dir.normalize();
        let side = up.cross(&dir).normalize();
        let up = dir.cross(&side).normalize();

        Matrix3::from_cols(side, up, dir).transpose()
    }

    /// Create a matrix from a rotation around the `x` axis (pitch).
    pub fn from_angle_x(theta: Rad<S>) -> Matrix3<S> {
        // http://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations
        let (s, c) = sin_cos(theta);
        Matrix3::new( one(),     zero(),    zero(),
                     zero(),  c.clone(), s.clone(),
                     zero(), -s.clone(), c.clone())
    }

    /// Create a matrix from a rotation around the `y` axis (yaw).
    pub fn from_angle_y(theta: Rad<S>) -> Matrix3<S> {
        // http://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations
        let (s, c) = sin_cos(theta);
        Matrix3::new(c.clone(), zero(), -s.clone(),
                        zero(),  one(),     zero(),
                     s.clone(), zero(),  c.clone())
    }

    /// Create a matrix from a rotation around the `z` axis (roll).
    pub fn from_angle_z(theta: Rad<S>) -> Matrix3<S> {
        // http://en.wikipedia.org/wiki/Rotation_matrix#Basic_rotations
        let (s, c) = sin_cos(theta);
        Matrix3::new( c.clone(), s.clone(), zero(),
                     -s.clone(), c.clone(), zero(),
                         zero(),    zero(),  one())
    }

    /// Create a matrix from a set of euler angles.
    ///
    /// # Parameters
    ///
    /// - `x`: the angular rotation around the `x` axis (pitch).
    /// - `y`: the angular rotation around the `y` axis (yaw).
    /// - `z`: the angular rotation around the `z` axis (roll).
    pub fn from_euler(x: Rad<S>, y: Rad<S>, z: Rad<S>) -> Matrix3<S> {
        // http://en.wikipedia.org/wiki/Rotation_matrix#General_rotations
        let (sx, cx) = sin_cos(x);
        let (sy, cy) = sin_cos(y);
        let (sz, cz) = sin_cos(z);

        Matrix3::new(                cy * cz,                 cy * sz,     -sy,
                     -cx * sz + sx * sy * cz,  cx * cz + sx * sy * sz, sx * cy,
                      sx * sz + cx * sy * cz, -sx * cz + cx * sy * sz, cx * cy)
    }

    /// Create a matrix from a rotation around an arbitrary axis
    pub fn from_axis_angle(axis: &Vector3<S>, angle: Rad<S>) -> Matrix3<S> {
        let (s, c) = sin_cos(angle);
        let _1subc = one::<S>() - c;

        Matrix3::new(_1subc * axis.x * axis.x + c,
                     _1subc * axis.x * axis.y + s * axis.z,
                     _1subc * axis.x * axis.z - s * axis.y,

                     _1subc * axis.x * axis.y - s * axis.z,
                     _1subc * axis.y * axis.y + c,
                     _1subc * axis.y * axis.z + s * axis.x,

                     _1subc * axis.x * axis.z + s * axis.y,
                     _1subc * axis.y * axis.z - s * axis.x,
                     _1subc * axis.z * axis.z + c)
    }
}

impl<S: BaseNum> Matrix4<S> {
    /// Create a new matrix, providing values for each index.
    #[inline]
    pub fn new(c0r0: S, c0r1: S, c0r2: S, c0r3: S,
               c1r0: S, c1r1: S, c1r2: S, c1r3: S,
               c2r0: S, c2r1: S, c2r2: S, c2r3: S,
               c3r0: S, c3r1: S, c3r2: S, c3r3: S) -> Matrix4<S>  {
        Matrix4::from_cols(Vector4::new(c0r0, c0r1, c0r2, c0r3),
                           Vector4::new(c1r0, c1r1, c1r2, c1r3),
                           Vector4::new(c2r0, c2r1, c2r2, c2r3),
                           Vector4::new(c3r0, c3r1, c3r2, c3r3))
    }

    /// Create a new matrix, providing columns.
    #[inline]
    pub fn from_cols(c0: Vector4<S>, c1: Vector4<S>, c2: Vector4<S>, c3: Vector4<S>) -> Matrix4<S> {
        Matrix4 { x: c0, y: c1, z: c2, w: c3 }
    }

    /// Create a new diagonal matrix, providing a single value to use for each
    /// non-zero index.
    #[inline]
    pub fn from_value(value: S) -> Matrix4<S> {
        Matrix4::new(value.clone(),        zero(),        zero(),        zero(),
                            zero(), value.clone(),        zero(),        zero(),
                            zero(),        zero(), value.clone(),        zero(),
                            zero(),        zero(),        zero(), value.clone())
    }

    /// Create a zero matrix (all zeros).
    #[inline]
    pub fn zero() -> Matrix4<S> {
        Matrix4::from_value(zero())
    }

    /// Create an identity matrix (diagonal matrix of ones).
    #[inline]
    pub fn identity() -> Matrix4<S> {
        Matrix4::from_value(one())
    }

    /// Create a translation matrix from a Vector3
    #[inline]
    pub fn from_translation(v: &Vector3<S>) -> Matrix4<S> {
        Matrix4::new(one(),  zero(), zero(), zero(),
                     zero(), one(),  zero(), zero(),
                     zero(), zero(), one(),  zero(),
                     v.x,    v.y,    v.z,    one())
    }
}

impl<S: BaseFloat>
Matrix4<S> {
    /// Create a transformation matrix that will cause a vector to point at
    /// `dir`, using `up` for orientation.
    pub fn look_at(eye: &Point3<S>, center: &Point3<S>, up: &Vector3<S>) -> Matrix4<S> {
        let f = center.sub_p(eye).normalize();
        let s = f.cross(up).normalize();
        let u = s.cross(&f);

        Matrix4::new( s.x.clone(),  u.x.clone(), -f.x.clone(), zero(),
                      s.y.clone(),  u.y.clone(), -f.y.clone(), zero(),
                      s.z.clone(),  u.z.clone(), -f.z.clone(), zero(),
                     -eye.dot(&s), -eye.dot(&u),  eye.dot(&f),  one())
    }
}

pub trait Matrix<S: BaseFloat, V: Clone + Vector<S>>: Array2<V, V, S>
                                                    + Neg<Self>
                                                    + Zero + One
                                                    + ApproxEq<S> {
    /// Multiply this matrix by a scalar, returning the new matrix.
    fn mul_s(&self, s: S) -> Self;
    /// Divide this matrix by a scalar, returning the new matrix.
    fn div_s(&self, s: S) -> Self;
    /// Take the remainder of this matrix by a scalar, returning the new
    /// matrix.
    fn rem_s(&self, s: S) -> Self;

    /// Add this matrix with another matrix, returning the new metrix.
    fn add_m(&self, m: &Self) -> Self;
    /// Subtract another matrix from this matrix, returning the new matrix.
    fn sub_m(&self, m: &Self) -> Self;

    /// Multiplay a vector by this matrix, returning a new vector.
    fn mul_v(&self, v: &V) -> V;

    /// Multiply this matrix by another matrix, returning the new matrix.
    fn mul_m(&self, m: &Self) -> Self;

    /// Negate this matrix in-place (multiply by scalar -1).
    fn neg_self(&mut self);

    /// Multiply this matrix by a scalar, in-place.
    fn mul_self_s(&mut self, s: S);
    /// Divide this matrix by a scalar, in-place.
    fn div_self_s(&mut self, s: S);
    /// Take the remainder of this matrix, in-place.
    fn rem_self_s(&mut self, s: S);

    /// Add this matrix with another matrix, in-place.
    fn add_self_m(&mut self, m: &Self);
    /// Subtract another matrix from this matrix, in-place.
    fn sub_self_m(&mut self, m: &Self);

    /// Multiply this matrix by another matrix, in-place.
    #[inline]
    fn mul_self_m(&mut self, m: &Self) { *self = self.mul_m(m); }

    /// Transpose this matrix, returning a new matrix.
    fn transpose(&self) -> Self;
    /// Transpose this matrix in-place.
    fn transpose_self(&mut self);
    /// Take the determinant of this matrix.
    fn determinant(&self) -> S;

    /// Return a vector containing the diagonal of this matrix.
    fn diagonal(&self) -> V;

    /// Return the trace of this matrix. That is, the sum of the diagonal.
    #[inline]
    fn trace(&self) -> S { self.diagonal().comp_add() }

    /// Invert this matrix, returning a new matrix. `m.mul_m(m.invert())` is
    /// the identity matrix. Returns `None` if this matrix is not invertible
    /// (has a determinant of zero).
    fn invert(&self) -> Option<Self>;

    /// Invert this matrix in-place.
    #[inline]
    fn invert_self(&mut self) {
        *self = self.invert().expect("Attempted to invert a matrix with zero determinant.");
    }

    /// Test if this matrix is invertible.
    #[inline]
    fn is_invertible(&self) -> bool { !self.determinant().approx_eq(&zero()) }

    /// Test if this matrix is the identity matrix. That is, it is diagonal
    /// and every element in the diagonal is one.
    #[inline]
    fn is_identity(&self) -> bool { self.approx_eq(&one()) }

    /// Test if this is a diagonal matrix. That is, every element outside of
    /// the diagonal is 0.
    fn is_diagonal(&self) -> bool;

    /// Test if this matrix is symmetric. That is, it is equal to its
    /// transpose.
    fn is_symmetric(&self) -> bool;
}

impl<S: BaseFloat> Add<Matrix2<S>, Matrix2<S>> for Matrix2<S> { #[inline] fn add(&self, other: &Matrix2<S>) -> Matrix2<S> { self.add_m(other) } }
impl<S: BaseFloat> Add<Matrix3<S>, Matrix3<S>> for Matrix3<S> { #[inline] fn add(&self, other: &Matrix3<S>) -> Matrix3<S> { self.add_m(other) } }
impl<S: BaseFloat> Add<Matrix4<S>, Matrix4<S>> for Matrix4<S> { #[inline] fn add(&self, other: &Matrix4<S>) -> Matrix4<S> { self.add_m(other) } }

impl<S: BaseFloat> Sub<Matrix2<S>, Matrix2<S>> for Matrix2<S> { #[inline] fn sub(&self, other: &Matrix2<S>) -> Matrix2<S> { self.sub_m(other) } }
impl<S: BaseFloat> Sub<Matrix3<S>, Matrix3<S>> for Matrix3<S> { #[inline] fn sub(&self, other: &Matrix3<S>) -> Matrix3<S> { self.sub_m(other) } }
impl<S: BaseFloat> Sub<Matrix4<S>, Matrix4<S>> for Matrix4<S> { #[inline] fn sub(&self, other: &Matrix4<S>) -> Matrix4<S> { self.sub_m(other) } }

impl<S: BaseFloat> Neg<Matrix2<S>> for Matrix2<S> { #[inline] fn neg(&self) -> Matrix2<S> { Matrix2::from_cols(self.c(0).neg(), self.c(1).neg()) } }
impl<S: BaseFloat> Neg<Matrix3<S>> for Matrix3<S> { #[inline] fn neg(&self) -> Matrix3<S> { Matrix3::from_cols(self.c(0).neg(), self.c(1).neg(), self.c(2).neg()) } }
impl<S: BaseFloat> Neg<Matrix4<S>> for Matrix4<S> { #[inline] fn neg(&self) -> Matrix4<S> { Matrix4::from_cols(self.c(0).neg(), self.c(1).neg(), self.c(2).neg(), self.c(3).neg()) } }

impl<S: BaseFloat> Zero for Matrix2<S> { #[inline] fn zero() -> Matrix2<S> { Matrix2::zero() } #[inline] fn is_zero(&self) -> bool { *self == zero() } }
impl<S: BaseFloat> Zero for Matrix3<S> { #[inline] fn zero() -> Matrix3<S> { Matrix3::zero() } #[inline] fn is_zero(&self) -> bool { *self == zero() } }
impl<S: BaseFloat> Zero for Matrix4<S> { #[inline] fn zero() -> Matrix4<S> { Matrix4::zero() } #[inline] fn is_zero(&self) -> bool { *self == zero() } }

impl<S: BaseFloat> Mul<Matrix2<S>, Matrix2<S>> for Matrix2<S> { #[inline] fn mul(&self, other: &Matrix2<S>) -> Matrix2<S> { self.mul_m(other) } }
impl<S: BaseFloat> Mul<Matrix3<S>, Matrix3<S>> for Matrix3<S> { #[inline] fn mul(&self, other: &Matrix3<S>) -> Matrix3<S> { self.mul_m(other) } }
impl<S: BaseFloat> Mul<Matrix4<S>, Matrix4<S>> for Matrix4<S> { #[inline] fn mul(&self, other: &Matrix4<S>) -> Matrix4<S> { self.mul_m(other) } }

impl<S: BaseFloat> One for Matrix2<S> { #[inline] fn one() -> Matrix2<S> { Matrix2::identity() } }
impl<S: BaseFloat> One for Matrix3<S> { #[inline] fn one() -> Matrix3<S> { Matrix3::identity() } }
impl<S: BaseFloat> One for Matrix4<S> { #[inline] fn one() -> Matrix4<S> { Matrix4::identity() } }

impl<S: Copy> Array2<Vector2<S>, Vector2<S>, S> for Matrix2<S> {
    #[inline]
    fn ptr<'a>(&'a self) -> &'a S { &self.x.x }

    #[inline]
    fn mut_ptr<'a>(&'a mut self) -> &'a mut S { &mut self.x.x }

    #[inline]
    fn c<'a>(&'a self, c: uint) -> &'a Vector2<S> {
        let slice: &'a [Vector2<S>, ..2] = unsafe { mem::transmute(self) };
        &'a slice[c]
    }

    #[inline]
    fn mut_c<'a>(&'a mut self, c: uint) -> &'a mut Vector2<S> {
        let slice: &'a mut [Vector2<S>, ..2] = unsafe { mem::transmute(self) };
        &'a mut slice[c]
    }

    #[inline]
    fn r(&self, r: uint) -> Vector2<S> {
        Vector2::new(self.cr(0, r),
                     self.cr(1, r))
    }

    #[inline]
    fn swap_r(&mut self, a: uint, b: uint) {
        self.mut_c(0).swap_i(a, b);
        self.mut_c(1).swap_i(a, b);
    }

    #[inline]
    fn map(&mut self, op: |Vector2<S>| -> Vector2<S>) -> Matrix2<S> {
        self.x = op(self.x);
        self.y = op(self.y);
        *self
    }
}

impl<S: Copy> Array2<Vector3<S>, Vector3<S>, S> for Matrix3<S> {
    #[inline]
    fn ptr<'a>(&'a self) -> &'a S { &self.x.x }

    #[inline]
    fn mut_ptr<'a>(&'a mut self) -> &'a mut S { &mut self.x.x }

    #[inline]
    fn c<'a>(&'a self, c: uint) -> &'a Vector3<S> {
        let slice: &'a [Vector3<S>, ..3] = unsafe { mem::transmute(self) };
        &'a slice[c]
    }

    #[inline]
    fn mut_c<'a>(&'a mut self, c: uint) -> &'a mut Vector3<S> {
        let slice: &'a mut [Vector3<S>, ..3] = unsafe { mem::transmute(self) };
        &'a mut slice[c]
    }

    #[inline]
    fn r(&self, r: uint) -> Vector3<S> {
        Vector3::new(self.cr(0, r),
                     self.cr(1, r),
                     self.cr(2, r))
    }

    #[inline]
    fn swap_r(&mut self, a: uint, b: uint) {
        self.mut_c(0).swap_i(a, b);
        self.mut_c(1).swap_i(a, b);
        self.mut_c(2).swap_i(a, b);
    }

    #[inline]
    fn map(&mut self, op: |Vector3<S>| -> Vector3<S>) -> Matrix3<S> {
        self.x = op(self.x);
        self.y = op(self.y);
        self.z = op(self.z);
        *self
    }
}

impl<S: Copy> Array2<Vector4<S>, Vector4<S>, S> for Matrix4<S> {
    #[inline]
    fn ptr<'a>(&'a self) -> &'a S { &self.x.x }

    #[inline]
    fn mut_ptr<'a>(&'a mut self) -> &'a mut S { &mut self.x.x }

    #[inline]
    fn c<'a>(&'a self, c: uint) -> &'a Vector4<S> {
        let slice: &'a [Vector4<S>, ..4] = unsafe { mem::transmute(self) };
        &'a slice[c]
    }

    #[inline]
    fn mut_c<'a>(&'a mut self, c: uint) -> &'a mut Vector4<S> {
        let slice: &'a mut [Vector4<S>, ..4] = unsafe { mem::transmute(self) };
        &'a mut slice[c]
    }

    #[inline]
    fn r(&self, r: uint) -> Vector4<S> {
        Vector4::new(self.cr(0, r),
                     self.cr(1, r),
                     self.cr(2, r),
                     self.cr(3, r))
    }

    #[inline]
    fn swap_r(&mut self, a: uint, b: uint) {
        self.mut_c(0).swap_i(a, b);
        self.mut_c(1).swap_i(a, b);
        self.mut_c(2).swap_i(a, b);
        self.mut_c(3).swap_i(a, b);
    }

    #[inline]
    fn map(&mut self, op: |Vector4<S>| -> Vector4<S>) -> Matrix4<S> {
        self.x = op(self.x);
        self.y = op(self.y);
        self.z = op(self.z);
        self.w = op(self.w);
        *self
    }
}

impl<S: BaseFloat> Matrix<S, Vector2<S>> for Matrix2<S> {
    #[inline]
    fn mul_s(&self, s: S) -> Matrix2<S> {
        Matrix2::from_cols(self.c(0).mul_s(s),
                           self.c(1).mul_s(s))
    }

    #[inline]
    fn div_s(&self, s: S) -> Matrix2<S> {
        Matrix2::from_cols(self.c(0).div_s(s),
                           self.c(1).div_s(s))
    }

    #[inline]
    fn rem_s(&self, s: S) -> Matrix2<S> {
        Matrix2::from_cols(self.c(0).rem_s(s),
                           self.c(1).rem_s(s))
    }

    #[inline]
    fn add_m(&self, m: &Matrix2<S>) -> Matrix2<S> {
        Matrix2::from_cols(self.c(0).add_v(m.c(0)),
                           self.c(1).add_v(m.c(1)))
    }

    #[inline]
    fn sub_m(&self, m: &Matrix2<S>) -> Matrix2<S> {
        Matrix2::from_cols(self.c(0).sub_v(m.c(0)),
                           self.c(1).sub_v(m.c(1)))
    }

    #[inline]
    fn mul_v(&self, v: &Vector2<S>) -> Vector2<S> {
        Vector2::new(self.r(0).dot(v),
                     self.r(1).dot(v))
    }

    fn mul_m(&self, other: &Matrix2<S>) -> Matrix2<S> {
        Matrix2::new(self.r(0).dot(other.c(0)), self.r(1).dot(other.c(0)),
                     self.r(0).dot(other.c(1)), self.r(1).dot(other.c(1)))
    }

    #[inline]
    fn neg_self(&mut self) {
        self.mut_c(0).neg_self();
        self.mut_c(1).neg_self();
    }

    #[inline]
    fn mul_self_s(&mut self, s: S) {
        self.mut_c(0).mul_self_s(s);
        self.mut_c(1).mul_self_s(s);
    }

    #[inline]
    fn div_self_s(&mut self, s: S) {
        self.mut_c(0).div_self_s(s);
        self.mut_c(1).div_self_s(s);
    }

    #[inline]
    fn rem_self_s(&mut self, s: S) {
        self.mut_c(0).rem_self_s(s);
        self.mut_c(1).rem_self_s(s);
    }

    #[inline]
    fn add_self_m(&mut self, m: &Matrix2<S>) {
        self.mut_c(0).add_self_v(m.c(0));
        self.mut_c(1).add_self_v(m.c(1));
    }

    #[inline]
    fn sub_self_m(&mut self, m: &Matrix2<S>) {
        self.mut_c(0).sub_self_v(m.c(0));
        self.mut_c(1).sub_self_v(m.c(1));
    }

    fn transpose(&self) -> Matrix2<S> {
        Matrix2::new(self.cr(0, 0).clone(), self.cr(1, 0).clone(),
                     self.cr(0, 1).clone(), self.cr(1, 1).clone())
    }

    #[inline]
    fn transpose_self(&mut self) {
        self.swap_cr((0, 1), (1, 0));
    }

    #[inline]
    fn determinant(&self) -> S {
        self.cr(0, 0) * self.cr(1, 1) - self.cr(1, 0) * self.cr(0, 1)
    }

    #[inline]
    fn diagonal(&self) -> Vector2<S> {
        Vector2::new(self.cr(0, 0),
                     self.cr(1, 1))
    }

    #[inline]
    fn invert(&self) -> Option<Matrix2<S>> {
        let det = self.determinant();
        if det.approx_eq(&zero()) {
            None
        } else {
            Some(Matrix2::new( self.cr(1, 1) / det, -self.cr(0, 1) / det,
                              -self.cr(1, 0) / det,  self.cr(0, 0) / det))
        }
    }

    #[inline]
    fn is_diagonal(&self) -> bool {
        self.cr(0, 1).approx_eq(&zero()) &&
        self.cr(1, 0).approx_eq(&zero())
    }


    #[inline]
    fn is_symmetric(&self) -> bool {
        self.cr(0, 1).approx_eq(&self.cr(1, 0)) &&
        self.cr(1, 0).approx_eq(&self.cr(0, 1))
    }
}

impl<S: BaseFloat> Matrix<S, Vector3<S>> for Matrix3<S> {
    #[inline]
    fn mul_s(&self, s: S) -> Matrix3<S> {
        Matrix3::from_cols(self.c(0).mul_s(s),
                           self.c(1).mul_s(s),
                           self.c(2).mul_s(s))
    }

    #[inline]
    fn div_s(&self, s: S) -> Matrix3<S> {
        Matrix3::from_cols(self.c(0).div_s(s),
                           self.c(1).div_s(s),
                           self.c(2).div_s(s))
    }

    #[inline]
    fn rem_s(&self, s: S) -> Matrix3<S> {
        Matrix3::from_cols(self.c(0).rem_s(s),
                           self.c(1).rem_s(s),
                           self.c(2).rem_s(s))
    }

    #[inline]
    fn add_m(&self, m: &Matrix3<S>) -> Matrix3<S> {
        Matrix3::from_cols(self.c(0).add_v(m.c(0)),
                           self.c(1).add_v(m.c(1)),
                           self.c(2).add_v(m.c(2)))
    }

    #[inline]
    fn sub_m(&self, m: &Matrix3<S>) -> Matrix3<S> {
        Matrix3::from_cols(self.c(0).sub_v(m.c(0)),
                           self.c(1).sub_v(m.c(1)),
                           self.c(2).sub_v(m.c(2)))
    }

    #[inline]
    fn mul_v(&self, v: &Vector3<S>) -> Vector3<S> {
        Vector3::new(self.r(0).dot(v),
                     self.r(1).dot(v),
                     self.r(2).dot(v))
    }

    fn mul_m(&self, other: &Matrix3<S>) -> Matrix3<S> {
        Matrix3::new(self.r(0).dot(other.c(0)),self.r(1).dot(other.c(0)),self.r(2).dot(other.c(0)),
                     self.r(0).dot(other.c(1)),self.r(1).dot(other.c(1)),self.r(2).dot(other.c(1)),
                     self.r(0).dot(other.c(2)),self.r(1).dot(other.c(2)),self.r(2).dot(other.c(2)))
    }

    #[inline]
    fn neg_self(&mut self) {
        self.mut_c(0).neg_self();
        self.mut_c(1).neg_self();
        self.mut_c(2).neg_self();
    }

    #[inline]
    fn mul_self_s(&mut self, s: S) {
        self.mut_c(0).mul_self_s(s);
        self.mut_c(1).mul_self_s(s);
        self.mut_c(2).mul_self_s(s);
    }

    #[inline]
    fn div_self_s(&mut self, s: S) {
        self.mut_c(0).div_self_s(s);
        self.mut_c(1).div_self_s(s);
        self.mut_c(2).div_self_s(s);
    }

    #[inline]
    fn rem_self_s(&mut self, s: S) {
        self.mut_c(0).rem_self_s(s);
        self.mut_c(1).rem_self_s(s);
        self.mut_c(2).rem_self_s(s);
    }

    #[inline]
    fn add_self_m(&mut self, m: &Matrix3<S>) {
        self.mut_c(0).add_self_v(m.c(0));
        self.mut_c(1).add_self_v(m.c(1));
        self.mut_c(2).add_self_v(m.c(2));
    }

    #[inline]
    fn sub_self_m(&mut self, m: &Matrix3<S>) {
        self.mut_c(0).sub_self_v(m.c(0));
        self.mut_c(1).sub_self_v(m.c(1));
        self.mut_c(2).sub_self_v(m.c(2));
    }

    fn transpose(&self) -> Matrix3<S> {
        Matrix3::new(self.cr(0, 0).clone(), self.cr(1, 0).clone(), self.cr(2, 0).clone(),
                     self.cr(0, 1).clone(), self.cr(1, 1).clone(), self.cr(2, 1).clone(),
                     self.cr(0, 2).clone(), self.cr(1, 2).clone(), self.cr(2, 2).clone())
    }

    #[inline]
    fn transpose_self(&mut self) {
        self.swap_cr((0, 1), (1, 0));
        self.swap_cr((0, 2), (2, 0));
        self.swap_cr((1, 2), (2, 1));
    }

    fn determinant(&self) -> S {
        self.cr(0, 0) * (self.cr(1, 1) * self.cr(2, 2) - self.cr(2, 1) * self.cr(1, 2)) -
        self.cr(1, 0) * (self.cr(0, 1) * self.cr(2, 2) - self.cr(2, 1) * self.cr(0, 2)) +
        self.cr(2, 0) * (self.cr(0, 1) * self.cr(1, 2) - self.cr(1, 1) * self.cr(0, 2))
    }

    #[inline]
    fn diagonal(&self) -> Vector3<S> {
        Vector3::new(self.cr(0, 0),
                     self.cr(1, 1),
                     self.cr(2, 2))
    }

    fn invert(&self) -> Option<Matrix3<S>> {
        let det = self.determinant();
        if det.approx_eq(&zero()) { None } else {
            Some(Matrix3::from_cols(self.c(1).cross(self.c(2)).div_s(det.clone()),
                                    self.c(2).cross(self.c(0)).div_s(det.clone()),
                                    self.c(0).cross(self.c(1)).div_s(det.clone())).transpose())
        }
    }

    fn is_diagonal(&self) -> bool {
        self.cr(0, 1).approx_eq(&zero()) &&
        self.cr(0, 2).approx_eq(&zero()) &&

        self.cr(1, 0).approx_eq(&zero()) &&
        self.cr(1, 2).approx_eq(&zero()) &&

        self.cr(2, 0).approx_eq(&zero()) &&
        self.cr(2, 1).approx_eq(&zero())
    }

    fn is_symmetric(&self) -> bool {
        self.cr(0, 1).approx_eq(&self.cr(1, 0)) &&
        self.cr(0, 2).approx_eq(&self.cr(2, 0)) &&

        self.cr(1, 0).approx_eq(&self.cr(0, 1)) &&
        self.cr(1, 2).approx_eq(&self.cr(2, 1)) &&

        self.cr(2, 0).approx_eq(&self.cr(0, 2)) &&
        self.cr(2, 1).approx_eq(&self.cr(1, 2))
    }
}

// Using self.r(0).dot(other.c(0)) like the other matrix multiplies
// causes the LLVM to miss identical loads and multiplies. This optimization
// causes the code to be auto vectorized properly increasing the performance
// around ~4 times.
macro_rules! dot_matrix4(
    ($A:expr, $B:expr, $I:expr, $J:expr) => (
        ($A.cr(0, $I)) * ($B.cr($J, 0)) +
        ($A.cr(1, $I)) * ($B.cr($J, 1)) +
        ($A.cr(2, $I)) * ($B.cr($J, 2)) +
        ($A.cr(3, $I)) * ($B.cr($J, 3))
))

impl<S: BaseFloat> Matrix<S, Vector4<S>> for Matrix4<S> {
    #[inline]
    fn mul_s(&self, s: S) -> Matrix4<S> {
        Matrix4::from_cols(self.c(0).mul_s(s),
                           self.c(1).mul_s(s),
                           self.c(2).mul_s(s),
                           self.c(3).mul_s(s))
    }

    #[inline]
    fn div_s(&self, s: S) -> Matrix4<S> {
        Matrix4::from_cols(self.c(0).div_s(s),
                           self.c(1).div_s(s),
                           self.c(2).div_s(s),
                           self.c(3).div_s(s))
    }

    #[inline]
    fn rem_s(&self, s: S) -> Matrix4<S> {
        Matrix4::from_cols(self.c(0).rem_s(s),
                           self.c(1).rem_s(s),
                           self.c(2).rem_s(s),
                           self.c(3).rem_s(s))
    }

    #[inline]
    fn add_m(&self, m: &Matrix4<S>) -> Matrix4<S> {
        Matrix4::from_cols(self.c(0).add_v(m.c(0)),
                           self.c(1).add_v(m.c(1)),
                           self.c(2).add_v(m.c(2)),
                           self.c(3).add_v(m.c(3)))
    }

    #[inline]
    fn sub_m(&self, m: &Matrix4<S>) -> Matrix4<S> {
        Matrix4::from_cols(self.c(0).sub_v(m.c(0)),
                           self.c(1).sub_v(m.c(1)),
                           self.c(2).sub_v(m.c(2)),
                           self.c(3).sub_v(m.c(3)))
    }

    #[inline]
    fn mul_v(&self, v: &Vector4<S>) -> Vector4<S> {
        Vector4::new(self.r(0).dot(v),
                     self.r(1).dot(v),
                     self.r(2).dot(v),
                     self.r(3).dot(v))
    }

    fn mul_m(&self, other: &Matrix4<S>) -> Matrix4<S> {
        Matrix4::new(dot_matrix4!(self, other, 0, 0), dot_matrix4!(self, other, 1, 0), dot_matrix4!(self, other, 2, 0), dot_matrix4!(self, other, 3, 0),
                     dot_matrix4!(self, other, 0, 1), dot_matrix4!(self, other, 1, 1), dot_matrix4!(self, other, 2, 1), dot_matrix4!(self, other, 3, 1),
                     dot_matrix4!(self, other, 0, 2), dot_matrix4!(self, other, 1, 2), dot_matrix4!(self, other, 2, 2), dot_matrix4!(self, other, 3, 2),
                     dot_matrix4!(self, other, 0, 3), dot_matrix4!(self, other, 1, 3), dot_matrix4!(self, other, 2, 3), dot_matrix4!(self, other, 3, 3))
    }

    #[inline]
    fn neg_self(&mut self) {
        self.mut_c(0).neg_self();
        self.mut_c(1).neg_self();
        self.mut_c(2).neg_self();
        self.mut_c(3).neg_self();
    }

    #[inline]
    fn mul_self_s(&mut self, s: S) {
        self.mut_c(0).mul_self_s(s);
        self.mut_c(1).mul_self_s(s);
        self.mut_c(2).mul_self_s(s);
        self.mut_c(3).mul_self_s(s);
    }

    #[inline]
    fn div_self_s(&mut self, s: S) {
        self.mut_c(0).div_self_s(s);
        self.mut_c(1).div_self_s(s);
        self.mut_c(2).div_self_s(s);
        self.mut_c(3).div_self_s(s);
    }

    #[inline]
    fn rem_self_s(&mut self, s: S) {
        self.mut_c(0).rem_self_s(s);
        self.mut_c(1).rem_self_s(s);
        self.mut_c(2).rem_self_s(s);
        self.mut_c(3).rem_self_s(s);
    }

    #[inline]
    fn add_self_m(&mut self, m: &Matrix4<S>) {
        self.mut_c(0).add_self_v(m.c(0));
        self.mut_c(1).add_self_v(m.c(1));
        self.mut_c(2).add_self_v(m.c(2));
        self.mut_c(3).add_self_v(m.c(3));
    }

    #[inline]
    fn sub_self_m(&mut self, m: &Matrix4<S>) {
        self.mut_c(0).sub_self_v(m.c(0));
        self.mut_c(1).sub_self_v(m.c(1));
        self.mut_c(2).sub_self_v(m.c(2));
        self.mut_c(3).sub_self_v(m.c(3));
    }

    fn transpose(&self) -> Matrix4<S> {
        Matrix4::new(self.cr(0, 0).clone(), self.cr(1, 0).clone(), self.cr(2, 0).clone(), self.cr(3, 0).clone(),
                     self.cr(0, 1).clone(), self.cr(1, 1).clone(), self.cr(2, 1).clone(), self.cr(3, 1).clone(),
                     self.cr(0, 2).clone(), self.cr(1, 2).clone(), self.cr(2, 2).clone(), self.cr(3, 2).clone(),
                     self.cr(0, 3).clone(), self.cr(1, 3).clone(), self.cr(2, 3).clone(), self.cr(3, 3).clone())
    }

    fn transpose_self(&mut self) {
        self.swap_cr((0, 1), (1, 0));
        self.swap_cr((0, 2), (2, 0));
        self.swap_cr((0, 3), (3, 0));
        self.swap_cr((1, 2), (2, 1));
        self.swap_cr((1, 3), (3, 1));
        self.swap_cr((2, 3), (3, 2));
    }

    fn determinant(&self) -> S {
        let m0 = Matrix3::new(self.cr(1, 1).clone(), self.cr(2, 1).clone(), self.cr(3, 1).clone(),
                              self.cr(1, 2).clone(), self.cr(2, 2).clone(), self.cr(3, 2).clone(),
                              self.cr(1, 3).clone(), self.cr(2, 3).clone(), self.cr(3, 3).clone());
        let m1 = Matrix3::new(self.cr(0, 1).clone(), self.cr(2, 1).clone(), self.cr(3, 1).clone(),
                              self.cr(0, 2).clone(), self.cr(2, 2).clone(), self.cr(3, 2).clone(),
                              self.cr(0, 3).clone(), self.cr(2, 3).clone(), self.cr(3, 3).clone());
        let m2 = Matrix3::new(self.cr(0, 1).clone(), self.cr(1, 1).clone(), self.cr(3, 1).clone(),
                              self.cr(0, 2).clone(), self.cr(1, 2).clone(), self.cr(3, 2).clone(),
                              self.cr(0, 3).clone(), self.cr(1, 3).clone(), self.cr(3, 3).clone());
        let m3 = Matrix3::new(self.cr(0, 1).clone(), self.cr(1, 1).clone(), self.cr(2, 1).clone(),
                              self.cr(0, 2).clone(), self.cr(1, 2).clone(), self.cr(2, 2).clone(),
                              self.cr(0, 3).clone(), self.cr(1, 3).clone(), self.cr(2, 3).clone());

        self.cr(0, 0) * m0.determinant() -
        self.cr(1, 0) * m1.determinant() +
        self.cr(2, 0) * m2.determinant() -
        self.cr(3, 0) * m3.determinant()
    }

    #[inline]
    fn diagonal(&self) -> Vector4<S> {
        Vector4::new(self.cr(0, 0),
                     self.cr(1, 1),
                     self.cr(2, 2),
                     self.cr(3, 3))
    }

    fn invert(&self) -> Option<Matrix4<S>> {
        let det = self.determinant();
        if !det.approx_eq(&zero()) {
            let one: S = one();
            let inv_det = one / det;
            let t = self.transpose();
            let cf = |i, j| {
                let mat = match i {
                    0 => Matrix3::from_cols(t.y.truncate_n(j),
                                            t.z.truncate_n(j),
                                            t.w.truncate_n(j)),
                    1 => Matrix3::from_cols(t.x.truncate_n(j),
                                            t.z.truncate_n(j),
                                            t.w.truncate_n(j)),
                    2 => Matrix3::from_cols(t.x.truncate_n(j),
                                            t.y.truncate_n(j),
                                            t.w.truncate_n(j)),
                    3 => Matrix3::from_cols(t.x.truncate_n(j),
                                            t.y.truncate_n(j),
                                            t.z.truncate_n(j)),
                    _ => fail!("out of range")
                };
                let sign = if (i+j) & 1 == 1 {-one} else {one};
                mat.determinant() * sign * inv_det
            };

            Some(Matrix4::new(cf(0, 0), cf(0, 1), cf(0, 2), cf(0, 3),
                              cf(1, 0), cf(1, 1), cf(1, 2), cf(1, 3),
                              cf(2, 0), cf(2, 1), cf(2, 2), cf(2, 3),
                              cf(3, 0), cf(3, 1), cf(3, 2), cf(3, 3)))

        } else {
            None
        }
    }

    fn is_diagonal(&self) -> bool {
        self.cr(0, 1).approx_eq(&zero()) &&
        self.cr(0, 2).approx_eq(&zero()) &&
        self.cr(0, 3).approx_eq(&zero()) &&

        self.cr(1, 0).approx_eq(&zero()) &&
        self.cr(1, 2).approx_eq(&zero()) &&
        self.cr(1, 3).approx_eq(&zero()) &&

        self.cr(2, 0).approx_eq(&zero()) &&
        self.cr(2, 1).approx_eq(&zero()) &&
        self.cr(2, 3).approx_eq(&zero()) &&

        self.cr(3, 0).approx_eq(&zero()) &&
        self.cr(3, 1).approx_eq(&zero()) &&
        self.cr(3, 2).approx_eq(&zero())
    }

    fn is_symmetric(&self) -> bool {
        self.cr(0, 1).approx_eq(&self.cr(1, 0)) &&
        self.cr(0, 2).approx_eq(&self.cr(2, 0)) &&
        self.cr(0, 3).approx_eq(&self.cr(3, 0)) &&

        self.cr(1, 0).approx_eq(&self.cr(0, 1)) &&
        self.cr(1, 2).approx_eq(&self.cr(2, 1)) &&
        self.cr(1, 3).approx_eq(&self.cr(3, 1)) &&

        self.cr(2, 0).approx_eq(&self.cr(0, 2)) &&
        self.cr(2, 1).approx_eq(&self.cr(1, 2)) &&
        self.cr(2, 3).approx_eq(&self.cr(3, 2)) &&

        self.cr(3, 0).approx_eq(&self.cr(0, 3)) &&
        self.cr(3, 1).approx_eq(&self.cr(1, 3)) &&
        self.cr(3, 2).approx_eq(&self.cr(2, 3))
    }
}

impl<S: BaseFloat> ApproxEq<S> for Matrix2<S> {
    #[inline]
    fn approx_eq_eps(&self, other: &Matrix2<S>, epsilon: &S) -> bool {
        self.c(0).approx_eq_eps(other.c(0), epsilon) &&
        self.c(1).approx_eq_eps(other.c(1), epsilon)
    }
}

impl<S: BaseFloat> ApproxEq<S> for Matrix3<S> {
    #[inline]
    fn approx_eq_eps(&self, other: &Matrix3<S>, epsilon: &S) -> bool {
        self.c(0).approx_eq_eps(other.c(0), epsilon) &&
        self.c(1).approx_eq_eps(other.c(1), epsilon) &&
        self.c(2).approx_eq_eps(other.c(2), epsilon)
    }
}

impl<S: BaseFloat> ApproxEq<S> for Matrix4<S> {
    #[inline]
    fn approx_eq_eps(&self, other: &Matrix4<S>, epsilon: &S) -> bool {
        self.c(0).approx_eq_eps(other.c(0), epsilon) &&
        self.c(1).approx_eq_eps(other.c(1), epsilon) &&
        self.c(2).approx_eq_eps(other.c(2), epsilon) &&
        self.c(3).approx_eq_eps(other.c(3), epsilon)
    }
}

// Conversion traits

/// Represents types which can be converted to a Matrix2
pub trait ToMatrix2<S: BaseNum> {
    /// Convert this value to a Matrix2
    fn to_matrix2(&self) -> Matrix2<S>;
}

/// Represents types which can be converted to a Matrix3
pub trait ToMatrix3<S: BaseNum> {
    /// Convert this value to a Matrix3
    fn to_matrix3(&self) -> Matrix3<S>;
}

/// Represents types which can be converted to a Matrix4
pub trait ToMatrix4<S: BaseNum> {
    /// Convert this value to a Matrix4
    fn to_matrix4(&self) -> Matrix4<S>;
}

impl<S: BaseFloat> ToMatrix3<S> for Matrix2<S> {
    /// Clone the elements of a 2-dimensional matrix into the top-left corner
    /// of a 3-dimensional identity matrix.
    fn to_matrix3(&self) -> Matrix3<S> {
        Matrix3::new(self.cr(0, 0).clone(), self.cr(0, 1).clone(), zero(),
                     self.cr(1, 0).clone(), self.cr(1, 1).clone(), zero(),
                                    zero(),                zero(),  one())
    }
}

impl<S: BaseFloat> ToMatrix4<S> for Matrix2<S> {
    /// Clone the elements of a 2-dimensional matrix into the top-left corner
    /// of a 4-dimensional identity matrix.
    fn to_matrix4(&self) -> Matrix4<S> {
        Matrix4::new(self.cr(0, 0).clone(), self.cr(0, 1).clone(), zero(), zero(),
                     self.cr(1, 0).clone(), self.cr(1, 1).clone(), zero(), zero(),
                                    zero(),                zero(),  one(), zero(),
                                    zero(),                zero(), zero(),  one())
    }
}

impl<S: BaseFloat> ToMatrix4<S> for Matrix3<S> {
    /// Clone the elements of a 3-dimensional matrix into the top-left corner
    /// of a 4-dimensional identity matrix.
    fn to_matrix4(&self) -> Matrix4<S> {
        Matrix4::new(self.cr(0, 0).clone(), self.cr(0, 1).clone(), self.cr(0, 2).clone(), zero(),
                     self.cr(1, 0).clone(), self.cr(1, 1).clone(), self.cr(1, 2).clone(), zero(),
                     self.cr(2, 0).clone(), self.cr(2, 1).clone(), self.cr(2, 2).clone(), zero(),
                                    zero(),                zero(),                zero(),  one())
    }
}

impl<S: BaseFloat> ToQuaternion<S> for Matrix3<S> {
    /// Convert the matrix to a quaternion
    fn to_quaternion(&self) -> Quaternion<S> {
        // http://www.cs.ucr.edu/~vbz/resources/quatut.pdf
        let trace = self.trace();
        let half: S = cast(0.5f64).unwrap();
        match () {
            () if trace >= zero::<S>() => {
                let s = (one::<S>() + trace).sqrt();
                let w = half * s;
                let s = half / s;
                let x = (self.cr(1, 2) - self.cr(2, 1)) * s;
                let y = (self.cr(2, 0) - self.cr(0, 2)) * s;
                let z = (self.cr(0, 1) - self.cr(1, 0)) * s;
                Quaternion::new(w, x, y, z)
            }
            () if (self.cr(0, 0) > self.cr(1, 1)) && (self.cr(0, 0) > self.cr(2, 2)) => {
                let s = (half + (self.cr(0, 0) - self.cr(1, 1) - self.cr(2, 2))).sqrt();
                let w = half * s;
                let s = half / s;
                let x = (self.cr(0, 1) - self.cr(1, 0)) * s;
                let y = (self.cr(2, 0) - self.cr(0, 2)) * s;
                let z = (self.cr(1, 2) - self.cr(2, 1)) * s;
                Quaternion::new(w, x, y, z)
            }
            () if self.cr(1, 1) > self.cr(2, 2) => {
                let s = (half + (self.cr(1, 1) - self.cr(0, 0) - self.cr(2, 2))).sqrt();
                let w = half * s;
                let s = half / s;
                let x = (self.cr(0, 1) - self.cr(1, 0)) * s;
                let y = (self.cr(1, 2) - self.cr(2, 1)) * s;
                let z = (self.cr(2, 0) - self.cr(0, 2)) * s;
                Quaternion::new(w, x, y, z)
            }
            () => {
                let s = (half + (self.cr(2, 2) - self.cr(0, 0) - self.cr(1, 1))).sqrt();
                let w = half * s;
                let s = half / s;
                let x = (self.cr(2, 0) - self.cr(0, 2)) * s;
                let y = (self.cr(1, 2) - self.cr(2, 1)) * s;
                let z = (self.cr(0, 1) - self.cr(1, 0)) * s;
                Quaternion::new(w, x, y, z)
            }
        }
    }
}

impl<S: BaseNum> fmt::Show for Matrix2<S> {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "[[{}, {}], [{}, {}]]",
                self.cr(0, 0), self.cr(0, 1),
                self.cr(1, 0), self.cr(1, 1))
    }
}

impl<S: BaseNum> fmt::Show for Matrix3<S> {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "[[{}, {}, {}], [{}, {}, {}], [{}, {}, {}]]",
                self.cr(0, 0), self.cr(0, 1), self.cr(0, 2),
                self.cr(1, 0), self.cr(1, 1), self.cr(1, 2),
                self.cr(2, 0), self.cr(2, 1), self.cr(2, 2))
    }
}

impl<S: BaseNum> fmt::Show for Matrix4<S> {
    fn fmt(&self, f: &mut fmt::Formatter) -> fmt::Result {
        write!(f, "[[{}, {}, {}, {}], [{}, {}, {}, {}], [{}, {}, {}, {}], [{}, {}, {}, {}]]",
                self.cr(0, 0), self.cr(0, 1), self.cr(0, 2), self.cr(0, 3),
                self.cr(1, 0), self.cr(1, 1), self.cr(1, 2), self.cr(1, 3),
                self.cr(2, 0), self.cr(2, 1), self.cr(2, 2), self.cr(2, 3),
                self.cr(3, 0), self.cr(3, 1), self.cr(3, 2), self.cr(3, 3))
    }
}