Matrix Language

How to run

## From download
./matrix-lang
## From source
make run # compile and run
make main # compile
./main.out # run compiled

Defining variables

• Data types are fraction and matrix (which consists of fractions)
• The get_matrix function requires user input after the function call
• Everything is passed by value (no pointers)

>> x = -5√2/7
-5√2/7
>> y = -√2/7
-1√2/7
>> z = 10
10
>> a = 5/10
1/2
>> a = get_matrix(2, 2)
1/2 √2 1
3
1/2 √2 
1 3
>> a
1/2 √2 
1 3

Function calls

• Functions work similar to most other languages

function_name(parameter1, parameter2, ...)

Default Functions

get_matrix

• Description: Prompts the user to input a matrix with the specified number of rows and columns.
• Parameters:
  • rows (Fraction): The number of rows in the matrix.
  • cols (Fraction): The number of columns in the matrix.
• Returns: A matrix of the specified dimensions filled with user input.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> a
  1 2
  3 4
  

rref

• Description: Converts a matrix to its reduced row echelon form (RREF).
• Parameters:
  • matrix (Matrix): The matrix to be converted.
• Returns: The matrix in RREF.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> rref(a)
  1 0
  0 1
  

solve_equations

• Description: Solves a system of linear equations represented by a matrix.
• Parameters:
  • matrix (Matrix): The matrix representing the system of equations.
• Returns: None. Prints the solution to the console.
• Examples:
  • No solutions (x+2y=3, 4x+8y=6)

    >> a = get_matrix(2, 3)
    1 2 3
    4 8 6
    >> solve_equations(a)
    No solutions!
    

  • One solution (x+2y=3, 2x+4y=6 with a solution of x=1, y=1):

    >> a = get_matrix(2, 3)
    1 2 3
    4 5 9
    >> solve_equations(a)
    One solution!
    -------------
    1 1 
    -------------
    

  • Many solutions (x+2y=3, 2x+4y=6 with a solution of x=0+t, y=3/2-(1/2)t):

    >> a = get_matrix(2, 3)
    1 2 3
    2 4 6
    >> solve_equations(a)
    Multiple solutions!
    x0
    -------------
    1 -1/2 
    -------------
    Offset
    -------------
    0 3/2 
    ----------------
    

orthonormalize

• Description: Orthonormalizes the columns of a matrix using the Gram-Schmidt process.
• Parameters:
  • matrix (Matrix): The matrix to be orthonormalized.
• Returns: The orthonormalized matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> ortho(a)
  √10/10 3√10/10 
  3√10/10 -1√10/10 
  

qr

• Description: Computes the QR decomposition of a matrix (The orthogonal matrix Q can be retrieved from ortho(matrix)).
• Parameters:
  • matrix (Matrix): The matrix to be decomposed.
• Returns: The upper-triangular R matrix from the QR decomposition.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> qr(a)
  √10 7√10/5 
  0 √10/5
  >> ortho(a) * qr(a)
  1 2
  3 4
  

rows

• Description: Returns the number of rows in a matrix.
• Parameters:
  • matrix (Matrix): The matrix whose rows are to be counted.
• Returns: The number of rows in the matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> rows(a)
  2
  

cols

• Description: Returns the number of columns in a matrix.
• Parameters:
  • matrix (Matrix): The matrix whose columns are to be counted.
• Returns: The number of columns in the matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> cols(a)
  2
  

transpose

• Description: Transposes a matrix.
• Parameters:
  • matrix (Matrix): The matrix to be transposed.
• Returns: The transposed matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> transpose(a)
  1 3
  2 4
  

inverse

• Description: Computes the inverse of a matrix or a fraction.
• Parameters:
  • matrix (Matrix) or fraction (Fraction): The matrix or fraction to be inverted.
• Returns: The inverse of the matrix or fraction.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> inverse(a)
  -2 1
  1.5 -0.5
  

determinant

• Description: Computes the determinant of a matrix.
• Parameters:
  • matrix (Matrix): The matrix whose determinant is to be computed.
• Returns: The determinant of the matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> determinant(a)
  -2
  

norm_sq

• Description: Computes the squared norm of a matrix.
• Parameters:
  • matrix (Matrix): The matrix whose squared norm is to be computed.
• Returns: The squared norm of the matrix.
• Example:

  >> a = get_matrix(2, 2)
  1 2
  3 4
  >> norm_sq(a)
  30
  

Infix Operators

• The * operator between matrices uses matrix-matrix multiplciation
• The * operator between matrices and fractions uses scalar-matrix multiplication
• The only supported operators are + and * (to do division just inverse the second fraction and multiply)
• The * operator is prioritized over the + operator (unless parenthesis are used)

>> a = get_matrix(2, 2)
1 2 3 4
1 2
3 4
>> a * transpose(a)
5 11 
11 25 
>> (5/2 + 8/2) * a
13/2 13 
39/2 26 
>> 5/2 * inverse(8/2)
5/8