矩阵和向量的运算
提供一些概述和细节：关于矩阵、向量以及标量的运算。
Eigen提供了matrix/vector的运算操作，既包括重载了c++的算术运算符+/-/*，也引入了一些特殊的运算比如点乘dot、叉乘cross等。
对于Matrix类（matrix和vectors）这些操作只支持线性代数运算，比如：matrix1*matrix2表示矩阵的乘机，vetor+scalar是不允许的。如果你想执行非线性代数操作，请看下一篇（暂时放下）。
左右两侧变量具有相同的尺寸（行和列），并且元素类型相同（Eigen不自动转化类型）操作包括：
- 二元运算 + 如a+b
- 二元运算 - 如a-b
- 一元运算 - 如-a
- 复合运算 += 如a+=b
- 复合运算 -= 如a-=b
#include <iostream>#include <Eigen/Dense>using namespace Eigen;int main(){Matrix2d a;a << 1, 2,3, 4;MatrixXd b(2,2);b << 2, 3,1, 4;std::cout << "a + b =\n" << a + b << std::endl;std::cout << "a - b =\n" << a - b << std::endl;std::cout << "Doing a += b;" << std::endl;a += b;std::cout << "Now a =\n" << a << std::endl;Vector3d v(1,2,3);Vector3d w(1,0,0);std::cout << "-v + w - v =\n" << -v + w - v << std::endl;}输出：
a + b =
3 5
4 8
a - b =
-1 -1
2 0
Doing a += b;
Now a =
3 5
4 8
-v + w - v =
-1
-4
-6
乘/除标量是非常简单的，如下：
- 二元运算 * 如matrix*scalar
- 二元运算 * 如scalar*matrix
- 二元运算 / 如matrix/scalar
- 复合运算 *= 如matrix*=scalar
- 复合运算 /= 如matrix/=scalar
#include <iostream>#include <Eigen/Dense>using namespace Eigen;int main(){Matrix2d a;a << 1, 2,3, 4;Vector3d v(1,2,3);std::cout << "a * 2.5 =\n" << a * 2.5 << std::endl;std::cout << "0.1 * v =\n" << 0.1 * v << std::endl;std::cout << "Doing v *= 2;" << std::endl;v *= 2;std::cout << "Now v =\n" << v << std::endl;}结果
a * 2.5 =
2.5 5
7.5 10
0.1 * v =
0.1
0.2
0.3
Doing v *= 2;
Now v =
2
4
6
这里简单介绍，在高级主题中会详细解释。在Eigen中，线性运算比如+不会对变量自身做任何操作，会返回一个“表达式对象”来描述被执行的计算。当整个表达式被评估完（一般是遇到=号），实际的操作才执行。
这样做主要是为了优化，比如
VectorXf a(50), b(50), c(50), d(50);
...
a = 3*b + 4*c + 5*d;
Eigen会编译这段代码最终遍历一次即可运算完成。
for(int i = 0; i < 50; ++i)
a[i] = 3*b[i] + 4*c[i] + 5*d[i];
因此，我们不必要担心大的线性表达式的运算效率。
MatrixXcf a = MatrixXcf::Random(2,2);cout << "Here is the matrix a\n" << a << endl;cout << "Here is the matrix a^T\n" << a.transpose() << endl;cout << "Here is the conjugate of a\n" << a.conjugate() << endl;cout << "Here is the matrix a^*\n" << a.adjoint() << endl;输出
Here is the matrix a
(-0.211,0.68) (-0.605,0.823)
(0.597,0.566) (0.536,-0.33)
Here is the matrix a^T
(-0.211,0.68) (0.597,0.566)
(-0.605,0.823) (0.536,-0.33)
Here is the conjugate of a
(-0.211,-0.68) (-0.605,-0.823)
(0.597,-0.566) (0.536,0.33)
Here is the matrix a^*
(-0.211,-0.68) (0.597,-0.566)
(-0.605,-0.823) (0.536,0.33)
对于实数矩阵，conjugate不执行任何操作，adjoint等价于transpose。
transpose和adjoint会简单的返回一个代理对象并不对本省做转置。如果执行 b=a.transpose() ，a不变，转置结果被赋值给b。如果执行 a=a.transpose() Eigen在转置结束之前结果会开始写入a，所以a的最终结果不一定等于a的转置。
Matrix2i a; a << 1, 2, 3, 4;cout << "Here is the matrix a:\n" << a << endl;a = a.transpose(); // !!! do NOT do this !!!cout << "and the result of the aliasing effect:\n" << a << endl;
Here is the matrix a:
1 2
3 4
and the result of the aliasing effect:
1 2
2 4
这被称为“别名问题”。在debug模式，当assertions打开的情况加，这种常见陷阱可以被自动检测到。
对 a=a.transpose() 这种操作，可以执行in-palce转置。类似还有adjointInPlace。
MatrixXf a(2,3); a << 1, 2, 3, 4, 5, 6;cout << "Here is the initial matrix a:\n" << a << endl;a.transposeInPlace();cout << "and after being transposed:\n" << a << endl;
Here is the initial matrix a:
1 2 3
4 5 6
and after being transposed:
1 4
2 5
3 6
向量也是一种矩阵，实质都是矩阵-矩阵的乘法。
- 二元运算 *如a*b
- 复合运算 *=如a*=b
#include <iostream>#include <Eigen/Dense>using namespace Eigen;int main(){Matrix2d mat;mat << 1, 2,3, 4;Vector2d u(-1,1), v(2,0);std::cout << "Here is mat*mat:\n" << mat*mat << std::endl;std::cout << "Here is mat*u:\n" << mat*u << std::endl;std::cout << "Here is u^T*mat:\n" << u.transpose()*mat << std::endl;std::cout << "Here is u^T*v:\n" << u.transpose()*v << std::endl;std::cout << "Here is u*v^T:\n" << u*v.transpose() << std::endl;std::cout << "Let's multiply mat by itself" << std::endl;mat = mat*mat;std::cout << "Now mat is mat:\n" << mat << std::endl;}输出
Here is mat*mat:
7 10
15 22
Here is mat*u:
1
1
Here is u^T*mat:
2 2
Here is u^T*v:
-2
Here is u*v^T:
-2 -0
2 0
Let's multiply mat by itself
Now mat is mat:
7 10
15 22
m=m*m并不会导致别名问题，Eigen在这里做了特殊处理，引入了临时变量。实质将编译为：
tmp = m*m
m = tmp
如果你确定矩阵乘法是安全的（并没有别名问题），你可以使用noalias()函数来避免临时变量 c.noalias() += a*b 。
dot()执行点积，cross()执行叉积，点运算得到1*1的矩阵。当然，点运算也可以用u.adjoint()*v来代替。
#include <iostream>#include <Eigen/Dense>using namespace Eigen;using namespace std;int main(){Vector3d v(1,2,3);Vector3d w(0,1,2);cout << "Dot product: " << v.dot(w) << endl;double dp = v.adjoint()*w; // automatic conversion of the inner product to a scalarcout << "Dot product via a matrix product: " << dp << endl;cout << "Cross product:\n" << v.cross(w) << endl;}输出
Dot product: 8
Dot product via a matrix product: 8
Cross product:
1
-2
1
注意：点积只对三维vector有效。对于复数，Eigen的点积是第一个变量共轭和第二个变量的线性积。
Eigen提供了而一些归约函数：sum()、prod()、maxCoeff()和minCoeff()，他们对所有元素进行操作。
#include <iostream>#include <Eigen/Dense>using namespace std;int main(){Eigen::Matrix2d mat;mat << 1, 2,3, 4;cout << "Here is mat.sum(): " << mat.sum() << endl;cout << "Here is mat.prod(): " << mat.prod() << endl;cout << "Here is mat.mean(): " << mat.mean() << endl;cout << "Here is mat.minCoeff(): " << mat.minCoeff() << endl;cout << "Here is mat.maxCoeff(): " << mat.maxCoeff() << endl;cout << "Here is mat.trace(): " << mat.trace() << endl;}输出
Here is mat.sum(): 10
Here is mat.prod(): 24
Here is mat.mean(): 2.5
Here is mat.minCoeff(): 1
Here is mat.maxCoeff(): 4
Here is mat.trace(): 5
trace表示矩阵的迹，对角元素的和等价于 a.diagonal().sum() 。
minCoeff和maxCoeff函数也可以返回结果元素的位置信息。
Matrix3f m = Matrix3f::Random();std::ptrdiff_t i, j;float minOfM = m.minCoeff(&i,&j);cout << "Here is the matrix m:\n" << m << endl;cout << "Its minimum coefficient (" << minOfM<< ") is at position (" << i << "," << j << ")\n\n";RowVector4i v = RowVector4i::Random();int maxOfV = v.maxCoeff(&i);cout << "Here is the vector v: " << v << endl;cout << "Its maximum coefficient (" << maxOfV<< ") is at position " << i << endl;输出
Here is the matrix m:
0.68 0.597 -0.33
-0.211 0.823 0.536
0.566 -0.605 -0.444
Its minimum coefficient (-0.605) is at position (2,1)
Here is the vector v: 1 0 3 -3
Its maximum coefficient (3) is at position 2
Eigen会检测执行操作的有效性，在编译阶段Eigen会检测它们，错误信息是繁冗的，但错误信息会大写字母突出，比如:
Matrix3f m;Vector4f v;v = m*v; // Compile-time error: YOU_MIXED_MATRICES_OF_DIFFERENT_SIZES当然动态尺寸的错误要在运行时发现，如果在debug模式，assertions会触发后，程序将崩溃。
MatrixXf m(3,3);VectorXf v(4);v = m * v; // Run-time assertion failure here: "invalid matrix product"
