math/smatrix/doc/Functions.md

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// 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:

<pre>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;
</pre>

### 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$:

<pre>_// M is a  N1xN2 matrix, x is a N2 size vector, y is a N1 size vector_
y = M * x
</pre>

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$:

<pre>_// A is a N1xN2 matrix, B is a N2xN3 matrix and C is a N1xN3 matrix_
C = A * B
</pre>

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:

*   **ROOT::Math::Transpose**(M) : return the transpose matrix, \f$ M^T \f$
*   **ROOT::Math::Similarity**( v, M) : returns the scalar value resulting from the matrix- vector product \f$ v^T M v \f$
*   **ROOT::Math::Similarity**( U, M) : returns the matrix resulting from the product \f$ U M U^T \f$. If M is symmetric, the returned resulting matrix is also symmetric
*   **ROOT::Math::SimilarityT**( U, M) : returns the matrix resulting from the product \f$ U^T M U \f$. If M is symmetric, the returned resulting matrix is also symmetric

See \ref MatrixFunctions for the documentation of all existing matrix functions in the package.   
The major Vector functions are:

*   **ROOT::Math::Dot**( v1, v2) : returns the scalar value resulting from the vector dot product
*   **ROOT::Math::Cross**( v1, v2) : returns the vector cross product for two vectors of size 3\. Note that the Cross product is not defined for other vector sizes
*   **ROOT::Math::Unit**( v) : returns unit vector. One can use also the _v.Unit()_ method.
*   **ROOT::Math::TensorProd**(v1,v2) : returns a general matrix M of size N1xN2 resulting from the [Tensor Product](http://en.wikipedia.org/wiki/Tensor_product) between the vector v1 of size N1) and v2 of size N2

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:

<pre>v.Print(std::cout);
std::cout << v << std::endl;
</pre>

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.

*/

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