// Matrix and Vector Functions doc /** \page MatVecFunctions Matrix and Vector Operators and Functions

Matrix and Vector Operators

The ROOT::Math::SVector and ROOT::Math::SMatrix classes defines the following operators described below. The m1,m2,m3 are vectors or matrices of the same type (and size) and a is a scalar value:

m1 == m2           // returns whether m1 is equal to m2 (element by element comparison)  
m1 != m2           // returns whether m1 is NOT equal to m2 (element by element comparison)  
m1 < m2            // returns whether m1 is less than m2 (element wise comparison)
m1 > m2            // returns whether m1 is greater than m2 (element wise comparison)
// in the following m1 and m3 can be general and m2 symmetric, but not vice-versa
m1 += m2           // add m2 to m1
m1 -= m2           // subtract m2 to m1 
m3 = m1 + m2       // addition 
m1 - m2            // subtraction

// Multiplication and division via a scalar value a 
m3 = a*m1; m3 = m1*a; m3 = m1/a;  

Vector-Vector multiplication

The operator * defines an element by element multiplication between vectors. For the standard vector-vector multiplication, \f$ a = v^T v \f$, (dot product) one must use the ROOT::Math::Dot function. In addition, the Cross (only for vector sizes of 3), ROOT::Math::Cross, and the Tensor product, ROOT::Math::TensorProd, are defined.

Matrix - Vector multiplication

The operator * defines the matrix-vector multiplication, \f$ y_i = \sum_{j} M_{ij} x_j\f$:
// M is a  N1xN2 matrix, x is a N2 size vector, y is a N1 size vector
y = M * x        
It compiles only if the matrix and the vectors have the right sizes.
Matrix - Matrix multiplication The operator * defines the matrix-matrix multiplication, \f$ C_{ij} = \sum_{k} A_{ik} B_{kj}\f$:
// A is a N1xN2 matrix, B is a N2xN3 matrix and C is a N1xN3 matrix
C = A * B    
The operation compiles only if the matrices have the right size. In the case that A and B are symmetric matrices, C is a general one, since their product is not guaranteed to be symmetric.

Matrix and Vector Functions

The most used matrix functions are: See \ref MatrixFunctions for the documentation of all existing matrix functions in the package.
The major Vector functions are: See \ref VectFunction for the list and documentation of all of them.

Matrix and Vector I/O

One can print (or write in an output stream) Vectors (and also Matrices) using the Print method or the << operator, like:
std::cout << v << std::endl;
In the ROOT distribution, the CINT dictionary is generated for SMatrix and SVector for double types and sizes up to 5. This allows the storage of them in a ROOT file. */