# Matrices and vectors

Type vector is a special data type in MQL5, which enables operations with vectors. A vector is a one-dimensional array of type double. It is one of the fundamental concepts of linear algebra, which is used in many fields of science, including physics, geometry, and others. Vectors are used to solve systems of linear equations, in 3D graphics and in other application areas. Vectors can be added and multiplied. The length or distance between vectors can be obtained through the Norm. In programming, vectors are usually represented by arrays of homogeneous elements, which may have no regular vector operations, i.e. when arrays cannot be added or multiply, and they have no norm.

Vectors can be represented as row vectors and string vectors when working with matrices. Also, vectors in linear algebra use the concepts of covariance and contravariance. These concepts do not make any difference when writing an MQL5 code, as only the programmer decides what each object of the vector type is. For example, it can be rotation, displacement or compression vector in 3D graphics.

Generally speaking, from the point of view of linear algebra, a number is also a vector, but in a one-dimensional vector space. A vector itself can be considered as a special case of a matrix.

Type matrix is another special data type in MQL5 to represent matrices. A matrix is actually a two-dimensional array of type double. Vectors and matrices have been introduced into MQL5 for easier operations with certain types of data sets. With them, developers can benefit from the linear algebra possibilities in a simple and math-like form. Matrices can be used to compactly write systems of linear or differential equations. The number of matrix rows corresponds to the number of equations, while the the number of columns is equal to the number of unknowns. As a result, systems of linear equations can be solved through matrix operations.

The following data types exist:

• matrix — a matrix containing double elements.
• matrixf — a matrix containing float elements.
• matrixc — a matrix containing complex elements.
• vector — a vector containing double elements.
• vectorf — a vector containing float elements.
• vectorc — a vector containing complex elements.

Template functions support notations like matrix<double>, matrix<float>, vector<double>, vector<float> instead of the corresponding types.

The following algebraic operations are defined for the matrices:

• Multiplication of suitable-size matrices: the number of columns in the left matrix must equal the number of rows in the right matrix
• Matrix multiplication by a column vector; multiplication of a row vector by a matrix according to the matrix multiplication rule. In this sense the vector is a special case of a matrix
• Matrix multiplication by a number, that is, by a scalar

Mathematics considers many different matrix types. For example, identity matrix, symmetric, skew-symmetric, upper and lower triangular matrices, and other types. Various Normal forms play an important role in the matrix theory. They represent a certain canonical form of a matrix, which can be obtained by means of certain transformations. In practice, normal forms that have additional properties, such as for example stability, are used.

The use of vectors and matrices, or rather, of special methods of the relevant types, enables the creation of simpler, briefer and clearer code, which is close to mathematical notation. With these methods, you can avoid the need to create nested loops or to mind correct indexing of arrays in calculations. Therefore, the use of these methods increases reliability and speed in developing complex programs.

## List of matrix and vector methods

Types matrix and vector include methods that correspond to the relevant NumPy library methods. Using these methods, you can translate algorithms and codes from Python to MQL5 with minimum efforts. A lot of data processing tasks, mathematical equations, neural networks and machine learning tasks can be solved using ready-made Python methods and libraries.

Method matrix/vector

Analogous method in NumPy

Description

void matrix.Eye(const int rows, const int cols, const int ndiag=0)

eye

Construct a matrix with ones on the diagonal and zeros elsewhere

void matrix.Identity(const int rows)

identity

Construct a square matrix with ones on the main diagonal

void matrix.Ones(const int rows, const int cols)

ones

Construct a new matrix of given rows and columns, filled with ones

void matrix.Zeros(const int rows, const int cols)

zeros

Construct a new matrix of given rows and columns, filled with zeros

void matrix.Full(const int rows, const int cols, const scalar value)

full

Construct a new matrix of given rows and columns, filled with scalar value

void matrix.Copy(const matrix a)

copy

Construct a copy of the given matrix

void matrix.FromBuffer(const int rows, const int cols, const scalar array[], const int count=-1, const int offset=0)

frombuffer

Construct a matrix created from a 1-dimensional array

void matrix.FromFile(const int rows, condt int cols, const int file_handle, const int count=-1, const int offset=0)

fromfile

Construct a matrix from data in a text or binary file

void vector.FromString(const string source, const string sep=" ")

fromstring

Construct a vector initialized from text data in a string

void vector.Arange(const scalar start, const scalar stop, const scalar step=1)

arange

Construct evenly spaced values within a given interval

void matrix.Diag(const vector v, const int ndiag=0)

diag

Extract a diagonal or construct a diagonal matrix

void matrix.Tri(const int rows, const int cols, const int ndiag=0)

tri

Construct a matrix with ones at and below the given diagonal and zeros elsewhere

void matrix.Tril(const int rows, const int cols, const scalar array[], const int ndiag=0)

tril

Return a copy of a matrix with elements above the k-th diagonal zeroed

void matrix.Triu(const int rows, const int cols, const scalar array[], const int ndiag=0)

triu

Return a copy of a matrix with the elements below the k-th diagonal zeroed

void matrix.Vander(const vector v, const int cols=-1, const bool increasing=false)

vander

Generate a Vandermonde matrix

vector matrix.Row(const unsigned nrow)

Return a row vector

vector matrix.Col(const unsigned ncol)

Return a column vector

unsigned matrix.Rows()

Return the number of rows in a matrix

unsigned matrix.Cols()

Return the number of columns in a matrix

void matrix.Init()

Initialize a matrix

matrix matrix.Transpose()

transpose

Reverse or permute the axes of a matrix; returns the modified matrix

matrix matrix.Dot(const matrix b)

dot

Dot product of two matrices

matrix matrix.Inner(const matrix b)

inner

Inner product of two matrices

matrix matrix.Outer(const matrix b)

outer

Compute the outer product of two matrices

matrix matrix.MatMul(const matrix b)

matmul

Matrix product of two matrices

matrix matrix.MatrixPower(const int power)

matrix_power

Raise a square matrix to the (integer) power n

matrix matrix.Kron(const matrix b)

kron

Return Kronecker product of two matrices

bool matrix.Cholesky(matrix& L)

cholesky

Return the Cholesky decomposition

bool matrix.QR(matrix& Q, matrix& R)

qr

Compute the qr factorization of a matrix

bool matrix.SVD(matrix& U, matrix& V, vector& singular_values)

svd

Singular value decomposition

bool matrix.Eig(matrix& eigen_vectors, vector& eigen_values)

eig

Compute the eigenvalues and right eigenvectors of a square matrix

bool matrix.EigH(matrix& eigen_vectors, vector& eigen_values)

eigh

Return the eigenvalues and eigenvectors of a Hermitian matrix

bool matrix.EigVals(vector& eigen_values)

eigvals

Compute the eigenvalues of a general matrix

bool matrix.EigValsH(vector& eigen_values)

eigvalsh

Compute the eigenvalues of a Hermitian matrix

bool matrix.LU(matrix& L, matrix& U)

LU decomposition of a matrix as the product of a lower triangular matrix and an upper triangular matrix

bool matrix.LUP(matrix& L, matrix& U, matrix& P)

LUP decomposition with partial pivoting, which refers to LU decomposition with row permutations only: PA=LU

double matrix.Norm(const norm)

norm

Return matrix or vector norm

double matrix.Cond(const norm)

cond

Compute the condition number of a matrix

vector matrix.Spectrum()

Compute spectrum of a matrix as the set of its eigenvalues from the product AT*A

double matrix.Det()

det

Compute the determinant of an array

int matrix.Rank()

matrix_rank

Return matrix rank of array using the Gaussian method

int matrix.SLogDet(int& sign)

slogdet

Compute the sign and logarithm of the determinant of an array

double matrix.Trace()

trace

Return the sum along diagonals of the matrix

vector matrix.Solve(const vector b)

solve

Solve a linear matrix equation, or system of linear algebraic equations

vector matrix.LstSq(const vector b)

lstsq

Return the least-squares solution of linear algebraic equations (for non-square or degenerate matrices)

matrix matrix.Inv()

inv

Compute the (multiplicative) inverse of a matrix

matrix matrix.PInv()

pinv

Compute the pseudo-inverse of a matrix by the Moore-Penrose method

int matrix.Compare(const matrix matrix_c, const double epsilon)

int matrix.Compare(const matrix matrix_c, const int digits)

int vector.Compare(const vector vector_c, const double epsilon)

int vector.Compare(const vector vector_c, const int digits)

Compare the elements of two matrices/vectors with the specified precision

double matrix.Flat(const ulong index)

bool matrix.Flat(const ulong index,const double value)

flat

double vector.ArgMax()

double matrix.ArgMax()

vector matrix.ArgMax(const int axis)

argmax

Return the index of the maximum value

double vector.ArgMin()

double matrix.ArgMin()

vector matrix.ArgMin(const int axis)

argmin

Return the index of the minimum value

double vector.Max()

double matrix.Max()

vector matrix.Max(const int axis)

max

Return the maximum value in a matrix/vector

double vector.Mean()

double matrix.Mean()

vector matrix.Mean(const int axis)

mean

Compute the arithmetic mean of element values

double vector.Min()

double matrix.Min()

vector matrix.Min(const int axis)

min

Return the minimum value in a matrix/vector

double vector.Sum()

double matrix.Sum()

vector matrix.Sum(const int axis)

sum

Return the sum of the matrix/vector elements which can also be performed for the given axis (axes).

void vector.Clip(const double min_value,const double max_value)

void matrix.Clip(const double min_value,const double max_value)

clip

Limit the elements of a matrix/vector to a specified range of valid values

vector vector.CumProd()

vector matrix.CumProd()

matrix matrix.CumProd(const int axis)

cumprod

Return the cumulative product of matrix/vector elements, including those along the given axis

vector vector.CumSum()

vector matrix.CumSum()

matrix matrix.CumSum(const int axis)

cumsum

Return the cumulative sum of matrix/vector elements, including those along the given axis

double vector.Prod(const double initial=1)

double matrix.Prod(const double initial=1)

vector matrix.Prod(const int axis,const double initial=1)

prod

Return the product of the matrix/vector elements which can also be performed for the given axis

void matrix.Reshape(const ulong rows, const ulong cols)

reshape

Change the shape of a matrix without changing its data

void matrix.Resize(const ulong rows,const ulong cols)

resize

Return a new matrix with a changed shape and size

bool matrix.SwapRows(const ulong row1, const ulong row2)

Swap rows in a matrix

bool matrix.SwapCols(const ulong col1, const ulong col2)

Swap columns in a matrix

double vector.Ptp()

double matrix.Ptp()

vector matrix.Ptp(const int axis)

ptp

Return the range of values of a matrix/vector or of the given matrix axis, equivalent to Max() - Min()

double vector.Percentile(const int percent)

double matrix.Percentile(const int percent)

vector matrix.Percentile(const int percent,const int axis)

percentile

Return the specified percentile of values of matrix/vector elements or elements along the specified axis. Valid values of the 'percent' parameter are in the range [0, 100]

double vector.Quantile(const int percent)

double matrix.Quantile(const int percent)

vector matrix.Quantile(const int percent,const int axis)

quantile

Return the specified quantile of values of matrix/vector elements or elements along the specified axis. The 'percent' parameter takes values in the range [0, 1]

double vector.Median()

double matrix.Median()

vector matrix.Median(const int axis)

median

Compute the median of the matrix/vector elements. The median is the middle value that separates the highest half of the array/vector elements from the lowest half of elements.

double vector.Average()

double matrix.Average()

vector matrix.Average(const int axis)

average

Compute the arithmetic mean of matrix/vector values. The sum of the weights in the denominator cannot be equal to 0, but some weights can be 0

double vector.Std()

double matrix.Std()

vector matrix.Std(const int axis)

std

Return the standard deviation of values of matrix/vector elements or of elements along the given axis

double vector.Var()

double matrix.Var()

vector matrix.Var(const int axis)

var

Compute the variance of values of matrix/vector elements

double vector.CorrCoef(const vector& v)

matrix matrix.CorrCoef()

corrcoef

Compute the Pearson correlation coefficient (linear correlation coefficient). The correlation coefficient is in the range [-1, 1]

vector vector.Correlate(const vector& v,enum mode)

correlate

Compute the cross-correlation of two vectors. The 'mode' parameter determines the linear convolution calculation mode

vector vector.Convolve(const vector& v, enum mode)

convolve

Return the discrete, linear convolution of two vectors The 'mode' parameter determines the linear convolution calculation mode

matrix matrix.Cov()

matrix vector.Cov(const vector& v); (resulting matrix 2 x 2)

cov

Compute the covariance matrix. The covariance of two samples (two random variables) is a measure of their linear dependence