====== Vectors and Matrices ======
The blocks in the control system are connected among each other with the aid of signals. As described in [[eeros_architecture:control_system:signals|]] they carry a timestamp and a value. The value could be a simple type such as an integer or float. Quite often it will consist of a vector or a matrix. A three dimensional vector of type ''double'' can be declared and initialized as follows:
<code cpp>
Vector<3, double> v; 
</code>
The declaration could be simplified to 
<code cpp>
Vector3<> v; 
</code>
as ''Vector3'' is a predefined type with three dimensions and the default element type is ''double''. A ''Vector3'' is identical to the type ''Matrix<3,1,double>''. The vector could be initialized as follows
<code cpp>
v = {1.5, -2, 0}
v << 1.5, -2, 0; // or with input operator
Vector3 v{1,2,3}; // or directly upon declaration
</code>
A matrix of 3 times 3 with element type ''int'' could be defined as
<code cpp>
Matrix<3, 3, int> m; 
m << 1, 2, 3,
     4, 5, 6,
     7, 8, 9;
</code>
The first three numbers will be filled into the first row. While the internal representation is simply a one dimensional vector, the matrix could be visualized as 

| ^col0^col1^col2^
^row0|1|2|3|
^row1|4|5|6|
^row2|7|8|9|
When declaring and initializing a matrix, the values must be given colon per colon. The same matrix as given above would be initialized as follows 
<code cpp>
Matrix<3, 3, int> m{1, 4, 7, 2, 5, 8, 3, 6, 9};
</code>  
===== Accessing Elements =====
You can access rows, columns or single elements of matrices with the following methods:
<code cpp>
m.get(0,0);  // returns element, 1
m.get(1,0);  // returns element, 4
m.getRow(1);  // returns row, [4,5,6]
m.getCol(0);  // returns column, [1,4,7]'
m.getSubMatrix<2,2>(0,1);   // returns matrix [[2,5][3,6]]
</code>
Single elements, rows or columns can be written with the methods //set()//, //setRow()//, or //setCol()//. 

The operators () and [] work as well as can be seen by the following example
<code cpp>
m(3);   // returns element or sets element with index 3 
m[3];   // returns element or sets element with index 3 
m(2,1);   // returns element or sets element with index 7
m[2,1];   // does not work, as the [] operator cannot be defined for two parameters
</code>

===== Logging Matrices =====
A matrix can be logged simply by writing

<code cpp>
log.info() << m;
</code>
With the 3 x 3 matrix from above we would get
<code>
2021-09-16 17:36:53:929    I:  [ [1 4 7]' [2 5 8]' [3 6 9]' ]
</code>
The matrix is plotted colon per colon. To give some more examples:
<code cpp>
Matrix<1,1> m1{1};
Matrix<2,1> m2{1,2};
Matrix<1,2> m3{1,2};
</code>
This matrices will be printed as 
<code>
2021-09-16 17:37:18:122    I:  [1]'
2021-09-16 17:37:18:122    I:  [1 2]'         // one colon with two rows
2021-09-16 17:37:18:122    I:  [ [1]' [2]' ]' // one row with two colons
</code>

===== Matrix Operations =====
Some examples show basic matrix operations.
<code cpp>
Vector2 v1{1,2};
Matrix<1,2> v2{3,4};
log.info() << v1 * v2;            // will print [ [3 6]' [4 8]' ]
log.info() << v2 * v1;            // will print [11]'

Matrix<2,2,double> m1{1,1.5,-1,2};
log.info() << v2 * m1;            // will print [ [9]' [5]' ]
log.info() << v1.transpose() * m1;// will print [ [4]' [3]' ]

auto x = v2 * m1;
Vector2 v4 = x.transpose();
log.info() << v4;                 // will print [9 5]'
</code>