matrix-c

Implementation of the matrix.h library.

Dependencies

• Work only in based UNIX system
• The make & check utility must be installed
• To format the code style you will need the clang-format option

Historical background

The first mentions of matrices (or as they were called then - "magic squares") were found in ancient China.
They became famous in the middle of the 18th century thanks to the work of the famous mathematician Gabriel Cramer, who published his work "Introduction to the Analysis of Algebraic Curves", which described a fundamentally new algorithm for solving systems of linear equations.
Soon after, the works of Carl Friedrich Gauss on the "classical" method of solving linear equations, the Cayley-Hamilton theorem, the works of Karl Weierstrass, Georg Frobenius, and other outstanding scientists were published.
It was not until 1850 that James Joseph Sylvester introduced the term "Matrix" in his work.

Matrix

A matrix is a collection of numbers arranged into a fixed number of rows and columns.

Matrix A is a rectangular table of numbers arranged in m rows and n columns

    1 2 3
A = 4 5 6
    7 8 9

     1  2  3  4
В =  5  6  7  8
     9 10 11 12

You can get the desired element with the help of indices, as follows A[1,1] = 1, where the first index is the row number, the second is the column number.

Matrix A will have elements with the following indices:

    (1,1) (1,2) (1,3)
A = (2,1) (2,2) (2,3)
    (3,1) (3,2) (3,3)

The order of a matrix is the number of its rows or columns.
The main diagonal of a square matrix is the diagonal from the upper left to the lower right corner.
A rectangular matrix (B) is a matrix with the number of rows not equal to the number of columns.
A square matrix (A) is a matrix with the number of rows equal to the number of columns.

A column matrix is a matrix with only one column:

    (1,1)
A = (2,1)
    (n,1)

A row matrix is a matrix that has only one row:

A = (1,1) (1,2) (1,m)

Tip: A column matrix and a row matrix are also often called vectors.

A diagonal matrix is a square matrix in which all elements outside the main diagonal are zero.
An identity matrix is a diagonal matrix with all diagonal elements equal to one:

    1 0 0
A = 0 1 0
    0 0 1

A triangular matrix is a square matrix with all elements on one side of the main diagonal equal to zero.

    1 2 3
A = 0 4 5
    0 0 6

Matrix operations

All operations (except matrix comparison) should return the resulting code:

• 0 - OK
• 1 - Error, incorrect matrix
• 2 - Calculation error (mismatched matrix sizes; matrix for which calculations cannot be performed, etc.)

Creating matrices (create_matrix)

int s21_create_matrix(int rows, int columns, matrix_t *result);

Cleaning of matrices (remove_matrix)

void s21_remove_matrix(matrix_t *A);

Matrix comparison (eq_matrix)

#define SUCCESS 1
#define FAILURE 0

int s21_eq_matrix(matrix_t *A, matrix_t *B);

The matrices A, B are equal |A = B| if they have the same dimensions and the corresponding elements are identical, thus for all i and j: A(i,j) = B(i,j)

The comparison must be up to and including 7 decimal places.

Adding (sum_matrix) and subtracting matrices (sub_matrix)

int s21_sum_matrix(matrix_t *A, matrix_t *B, matrix_t *result);
int s21_sub_matrix(matrix_t *A, matrix_t *B, matrix_t *result);

The sum of two matrices A = m × n and B = m × n of the same size is a matrix C = m × n = A + B of the same size whose elements are defined by the equations C(i,j) = A(i,j) + B(i,j).

The difference of two matrices A = m × n and B = m × n of the same size is a matrix C = m × n = A - B of the same size whose elements are defined by the equations C(i,j) = A(i,j) - B(i,j).

            1 2 3   1 0 0   2 2 3
С = A + B = 0 4 5 + 2 0 0 = 2 4 5
            0 0 6   3 4 1   3 4 7

Matrix multiplication by scalar (mult_number). Multiplication of two matrices (mult_matrix)

int s21_mult_number(matrix_t *A, double number, matrix_t *result);
int s21_mult_matrix(matrix_t *A, matrix_t *B, matrix_t *result);

The product of the matrix A = m × n by the number λ is the matrix B = m × n = λ × A whose elements are defined by the equations B = λ × A(i,j).

                1 2 3   2 4 6   
B = 2 × A = 2 × 0 4 2 = 0 8 4 
                2 3 4   4 6 8   

The product of A = m × k by B = k × n is a matrix C = m × n = A × B of size m × n whose elements are defined by the equation C(i,j) = A(i,1) × B(1,j) + A(i,2) × B(2,j) + ... + A(i,k) × B(k,j).

            1 4    1 -1  1    9 11 17   
C = A × B = 2 5  × 2  3  4 = 12 13 22
            3 6              15 15 27

The components of matrix C are calculated as follows:

C(1,1) = A(1,1) × B(1,1) + A(1,2) × B(2,1) = 1 × 1 + 4 × 2 = 1 + 8 = 9
C(1,2) = A(1,1) × B(1,2) + A(1,2) × B(2,2) = 1 × (-1) + 4 × 3 = (-1) + 12 = 11
C(1,3) = A(1,1) × B(1,3) + A(1,2) × B(2,3) = 1 × 1 + 4 × 4 = 1 + 16 = 17
C(2,1) = A(2,1) × B(1,1) + A(2,2) × B(2,1) = 2 × 1 + 5 × 2 = 2 + 10 = 12
C(2,2) = A(2,1) × B(1,2) + A(2,2) × B(2,2) = 2 × (-1) + 5 × 3 = (-2) + 15 = 13
C(2,3) = A(2,1) × B(1,3) + A(2,2) × B(2,3) = 2 × 1 + 5 × 4 = 2 + 20 = 22
C(3,1) = A(3,1) × B(1,1) + A(3,2) × B(2,1) = 3 × 1 + 6 × 2 = 3 + 12 = 15
C(3,2) = A(3,1) × B(1,2) + A(3,2) × B(2,2) = 3 × (-1) + 6 × 3 = (-3) + 18 = 15
C(3,3) = A(3,1) × B(1,3) + A(3,2) × B(2,3) = 3 × 1 + 6 × 4 = 3 + 24 = 27			

Matrix transpose (transpose)

int s21_transpose(matrix_t *A, matrix_t *result);

The transpose of matrix A is in switching its rows with its columns with their numbers retained

          1 4   1 2 3
A = A^T = 2 5 = 4 5 6
          3 6

Minor of matrix and matrix of algebraic complements (calc_complements)

int s21_calc_complements(matrix_t *A, matrix_t *result);

Minor M(i,j) is a (n-1)-order determinant obtained by deleting out the i-th row and the j-th column from the matrix A.

For the following matrix:

    1 2 3
A = 0 4 2
    5 2 1

The minor of the first element of the first row is:

M(1,1) = 4 2
         2 1

|M| = 4 - 4 = 0

The minors of matrix will look like this:

     0 -10 -20
M = -4 -14  -8
    -8   2   4

The algebraic complement of a matrix element is the value of the minor multiplied by -1^(i+j).

The matrix of algebraic complement will look like this:

      0  10 -20
M. =  4 -14   8
     -8  -2   4

Matrix determinant

int s21_determinant(matrix_t *A, double *result);

The determinant is a number that is associated to each square matrix and calculated from the elements using special formulas.
Tip: The determinant can only be calculated for a square matrix.

The determinant of a matrix equals the sum of the products of elements of the row (column) and the corresponding algebraic complements.

Finding the determinant of matrix A by the first row:

    1 2 3
A = 4 5 6
    7 8 9
	
|A| = 1 × 5 6 - 2 × 4 6 + 3 × 4 5 = 1 × (5 × 9 - 8 × 6) - 2 × (4 × 9 - 6 × 7) + 3 × (4 × 8 - 7 × 5)
          8 9       7 9       7 8
|A| = 1 × (45 - 48) - 2 × (36 - 42) + 3 × (32 - 35) = -3 + 12 + (-9) = 0
|A| = 0

Inverse of the matrix (inverse_matrix)

int s21_inverse_matrix(matrix_t *A, matrix_t *result);

A matrix A to the power of -1 is called the inverse of a square matrix A if the product of these matrices equals the identity matrix.

If the determinant of the matrix is zero, then it does not have an inverse.

The formula to calculate the inverse of matrix is $A^{-1}=\frac{1} {|A|} × A_*^T$

The following matrix is given:

     2  5  7
A =  6  3  4
     5 -2 -3

Finding the determinant:

|A| = -1

Determinant |A| != 0 -> matrix has an inverse.

Construction of minor matrix:

    -1 -38 -27
М = -1 -41 -29
    -1 -34 -24

The matrix of algebraic complements:

     -1  38 -27
М. =  1 -41  29
     -1  34 -24

The transpose of matrix of algebraic complements:

        -1   1  -1
М^T. =  38 -41  34
       -27  29 -24

The inverse matrix will look like this:

                           1  -1   1
A^(-1) =  1/|A| * M^T. = -38  41 -34
                          27 -29  24