Eigen之矩阵、向量、标量的操作运算
介绍
Eigen是通过中重载C++操作运算符如+、-、*或通过dot()、cross()等来实现矩阵/向量的操作运算
加法和减法
- binary operator + as in a+b
- binary operator - as in a-b
- unary operator - as in -a
- compound operator += as in a+=b
- compound operator -= as in a-=b
例子：
#include Output:
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标量的乘除
- binary operator * as in matrix*scalar
- binary operator * as in scalar*matrix
- binary operator / as in matrix/scalar
- compound operator = as in matrix=scalar
- compound operator /= as in matrix/=scalar
例子：
#include Output:
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
6A note about expression templates
这是我们在这个页面上解释的一个高级主题，但是现在只提到它是很有用的。在Eigen，算术运算符，比如运算符+不执行任何计算，他们只是返回一个描述计算的“表达式对象”。实际的计算发生在稍后，当整个表达式被求值时，通常在操作符=。虽然这听起来可能很重，但是任何现代的优化编译器都能够优化抽象，结果是完美的优化代码。例如，当你这样做的时候:
VectorXf a(50), b(50), c(50), d(50);
...
a = 3*b + 4*c + 5*d;Eigen将其编译为一个for循环，这样数组就只遍历一次。简化(例如忽略SIMD优化)，这个循环是这样的:
for(int i = 0; i < 50; ++i)
{
a[i] = 3*b[i] + 4*c[i] + 5*d[i];
}因此，你不应该害怕使用Eigen的相对大的算术表达式:这只给了Eigen更多的机会来优化。
矩阵的转置、共轭及伴随
例：
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;结果：
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)注：对于一个实数矩阵，conjugate()是没有进行什么操作的，因此adjoint() = transpose（）。
对于基本运算操作，transpose()和adjoint() 都仅是返回一个中间变量而没有进行实际的值改变。如果运算时b = a.transpose()，则结果值会赋值给b，但是如果进行的运算为a = a.transpose()，则Eigen会在转置完成之前将值赋值给a，因此无法达到期望的值。
例：
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;结果：
Here is the matrix a:
1 2
3 4
and the result of the aliasing effect:
1 2
2 4这种现象被称为混叠问题（aliasing issue）
为解决该问题，可以使用transposeInPlace()函数来完成：
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;结果：
Here is the initial matrix a:
1 2 3
4 5 6
and after being transposed:
1 4
2 5
3 6矩阵与矩阵、矩阵与向量间的乘法
- binary operator * as in a*b
- compound operator = as in a=b (this multiplies on the right: a*=b
is equivalent to a = a*b)
例：
#include 结果：
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提示：对于矩阵的乘法不会出现上述的混叠问题，因为Eigen将乘法作为一个特例
tmp = m*m;
m = tmp;点乘和叉乘
Eigen中通过dot()和cross()来实现点乘和叉乘
例：
#include 结果：
Dot product: 8
Dot product via a matrix product: 8
Cross product:
1
-2
1其他基础操作
Eigen还提供了一些函数实现一些其他基本操作，如sum()进行求和，prod()实现矩阵内值的积，maxCoeff()求矩阵内最大值等。
例：
#include 结果：
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(): 5minCoeff()函数和maxCoeff()函数的变换使用：
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;结果：
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【参考】http://eigen.tuxfamily.org/dox/group__TutorialMatrixArithmetic.html