Linear algebra support in commons-math provides operations on real matrices (both dense and sparse matrices are supported) and vectors. It features basic operations (addition, subtraction ...) and decomposition algorithms that can be used to solve linear systems either in exact sense and in least squares sense.

The RealMatrix interface represents a matrix with real numbers as entries. The following basic matrix operations are supported:

- Matrix addition, subtraction, multiplication
- Scalar addition and multiplication
- transpose
- Norm and Trace
- Operation on a vector

Example:

// Create a real matrix with two rows and three columns, using a factory // method that selects the implementation class for us. double[][] matrixData = { {1d,2d,3d}, {2d,5d,3d}}; RealMatrix m = MatrixUtils.createRealMatrix(matrixData); // One more with three rows, two columns, this time instantiating the // RealMatrix implementation class directly. double[][] matrixData2 = { {1d,2d}, {2d,5d}, {1d, 7d}}; RealMatrix n = new Array2DRowRealMatrix(matrixData2); // Note: The constructor copies the input double[][] array in both cases. // Now multiply m by n RealMatrix p = m.multiply(n); System.out.println(p.getRowDimension()); // 2 System.out.println(p.getColumnDimension()); // 2 // Invert p, using LU decomposition RealMatrix pInverse = new LUDecomposition(p).getSolver().getInverse();

The three main implementations of the interface are Array2DRowRealMatrix and BlockRealMatrix for dense matrices (the second one being more suited to dimensions above 50 or 100) and SparseRealMatrix for sparse matrices.

The RealVector interface represents a vector with real numbers as entries. The following basic matrix operations are supported:

- Vector addition, subtraction
- Element by element multiplication, division
- Scalar addition, subtraction, multiplication, division and power
- Mapping of mathematical functions (cos, sin ...)
- Dot product, outer product
- Distance and norm according to norms L1, L2 and Linf

The RealVectorFormat class handles input/output of vectors in a customizable textual format.

The `solve()` methods of the DecompositionSolver
interface support solving linear systems of equations of the form AX=B, either
in linear sense or in least square sense. A `RealMatrix` instance is
used to represent the coefficient matrix of the system. Solving the system is a
two phases process: first the coefficient matrix is decomposed in some way and
then a solver built from the decomposition solves the system. This allows to
compute the decomposition and build the solver only once if several systems have
to be solved with the same coefficient matrix.

For example, to solve the linear system

2x + 3y - 2z = 1 -x + 7y + 6x = -2 4x - 3y - 5z = 1

RealMatrix coefficients = new Array2DRowRealMatrix(new double[][] { { 2, 3, -2 }, { -1, 7, 6 }, { 4, -3, -5 } }, false); DecompositionSolver solver = new LUDecomposition(coefficients).getSolver();

RealVector constants = new ArrayRealVector(new double[] { 1, -2, 1 }, false); RealVector solution = solver.solve(constants);

Each type of decomposition has its specific semantics and constraints on the coefficient matrix as shown in the following table. For algorithms that solve AX=B in least squares sense the value returned for X is such that the residual AX-B has minimal norm. If an exact solution exist (i.e. if for some X the residual AX-B is exactly 0), then this exact solution is also the solution in least square sense. This implies that algorithms suited for least squares problems can also be used to solve exact problems, but the reverse is not true.

Decomposition algorithms | ||

Name | coefficients matrix | problem type |

LU | square | exact solution only |

Cholesky | symmetric positive definite | exact solution only |

QR | any | least squares solution |

eigen decomposition | square | exact solution only |

SVD | any | least squares solution |

It is possible to use a simple array of double instead of a `RealVector`.
In this case, the solution will be provided also as an array of double.

It is possible to solve multiple systems with the same coefficient matrix
in one method call. To do this, create a matrix whose column vectors correspond
to the constant vectors for the systems to be solved and use `solve(RealMatrix),`
which returns a matrix with column vectors representing the solutions.

Decomposition algorithms may be used for themselves and not only for linear system solving. This is of prime interest with eigen decomposition and singular value decomposition.

The `getEigenvalue()`, `getEigenvalues()`, `getEigenVector()`,
`getV()`, `getD()` and `getVT()` methods of the
`EigenDecomposition` interface support solving eigenproblems of the form
AX = lambda X where lambda is a real scalar.

The `getSingularValues()`, `getU()`, `getS()` and
`getV()` methods of the `SingularValueDecomposition` interface
allow to solve singular values problems of the form AXi = lambda Yi where lambda is a
real scalar, and where the Xi and Yi vectors form orthogonal bases of their respective
vector spaces (which may have different dimensions).

In addition to the real field, matrices and vectors using non-real field elements can be used. The fields already supported by the library are: