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Matrix multiplication

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Introduction Edit

Prerequisites Edit

It is assumed that those reading this have a basic understanding of what a matrix is and how to add them, and multiply them by scalars, i.e. plain old numbers like 3, or -5. A secondary school algebra course would probably give one more than enough background, but is surely not required by any means.

If you need some background Go here

Matrix Multiplication Basics Edit

In order to multiply 2 matrices given one must have the same amount of rows that the other has columns. In other words two matrices can be multiplied only if one is of dimension m×n and the other is of dimension n×p where m, n, and p are natural numbers {m,n,p \in \mathbb{N}}. The resulting matrix will be of dimension m×p.

Example 4x2 Multiplied by 2x4 Edit

Take a matrix 4x2 call it \mathbf{A}; another of dimension 2x4 and call it \mathbf{B}. Give them some arbitrary values and let's do some multiplication.poop

\mathbf{A} = \begin{bmatrix} 2 & 3 & -1 & 0 \\ -7 & 2 & 1 & 10 \end{bmatrix} \mathbf{B} = \begin{bmatrix} 3 & 4 \\ 2 & 1 \\ -1 & 2 \\ 2 & 7 \end{bmatrix}

Our result matrix is going to be 2×2 as you will see as we go step by step. First we multiply and sum the first row with the first column: 2(3) + 3(2) + (-1)(-1) + 0(2). Then we do the same for the first row and second column: 2(4) + 3(1) + (-1)(2)+ 0(7), etc.

To make a long story short, our matrix would be:

\mathbf{AB} = \begin{bmatrix} 2(3) + 3(2) + (-1)(-1) + 0(2) & 2(4) + 3(1) + (-1)(2) + 0(7) \\ -7(3) + 2(2) + 1 (-1) + 10(2) & -7(4) + 2(1) + 1 (2) + 10(7) \end{bmatrix} = \begin{bmatrix} 13 & 9 \\ 2 & 46 \end{bmatrix}

As this implies, multiplication is non-commutative in general for matrices, i.e. \mathbf{AB} \ne \mathbf{BA} since in this case if we reversed the order the resulting matrix \mathbf{BA}would be 4×4 instead of 2×2. 

More General Approach

Now let's visualize A and B as m×n and n×p matrices respectively.

\mathbf{A} = \begin{bmatrix} a_{1,1} & a_{1,2} & \dots & \dots \\ a_{2,1} & a_{2,2} & \dots & \dots \\ a_{3,1} & a_{3,2} & \ddots & \dots \\ \vdots & \vdots & \vdots & a_{m,n} \end{bmatrix} \mathbf{B} = \begin{bmatrix} b_{1,1} & b_{1,2} & \dots & \dots \\ b_{2,1} & b_{2,2} & \dots & \dots \\ b_{3,1} & b_{3,2} & \ddots & \dots \\ \vdots & \vdots & \vdots & b_{n,p} \end{bmatrix}

We are going to be adding and multiplying like before, but generally.

\mathbf{AB} = \begin{bmatrix} a_{1,1}b_{1,1} + a_{1,2}b_{2,1} + \dots + a_{1,n}b_{p,1} & \dots \\ \vdots & a_{m,1}b_{1,1} + a_{m,2}b_{2,p} + \dots + a_{m,n}b_{n,p} \end{bmatrix}

Pseudocode & General AlgorithmsEdit

So now that we have a general idea of what a matrix is, and how to multiply them in general, we can derive some pseudocode around it. We could break down the steps as follows.

  1. Check the sizes of two matrices \mathbf{A} (m×n) and \mathbf{B} (t×u): if n = t then we can multiply them otherwise no (in that order \mathbf{AB})
  2. If they can be multiplied, then create a new matrix of size m by u
  3. For each row in A and each column in \mathbf{B} multiply and sum the elements and the place the results in the rows and columns of the result matrix \mathbf{AB}

Here is some pseudocode treating matrices like if they have a m element and an n element, so the dimension of a matrix object is m×n. 


multiplyMatrix(matrix1, matrix2)

--  Multiplies rows and columns and sums them
  multiplyRowAndColumn(row, column) returns number
  var
    total: number
  begin
    for each rval in row and cval in column
    begin
       total += rval*cval
    end
    return total
  end

begin

--  If the rows don't match up then the function fails

  if matrix1:n != matrix2:m  return failure;

  dim    = matrix1:n   -- Could also be matrix2:m
  newmat = new squarematrix(dim)  -- Create a new dim x dim matrix
  for each r in matrix1:rows and c in matrix2:columns
  begin
    
  end
  
end

Implementations Edit

C Sharp Edit

// Program in C# to multiply two matrices using Rectangular arrays.
using System;
class MatrixMultiplication
{
	int[,] a;
	int[,] b;
	int[,] c;
	
	public void ReadMatrix()
	{
		Console.WriteLine("\n Size of Matrix 1:");
		Console.Write("\n Enter the number of rows in Matrix 1 :");
		int m=int.Parse(Console.ReadLine());
		Console.Write("\n Enter the number of columns in Matrix 1 :");
		int n=int.Parse(Console.ReadLine());
		a=new int[m,n];
		Console.WriteLine("\n Enter the elements of Matrix 1:");
		for(int i=0;i<a.GetLength(0);i++)
		{
			for(int j=0;j<a.GetLength(1);j++)
			{
				a[i,j]=int.Parse(Console.ReadLine());
			}
		}
		
		Console.WriteLine("\n Size of Matrix 2 :");
		Console.Write("\n Enter the number of rows in Matrix 2 :");
		m=int.Parse(Console.ReadLine());
		Console.Write("\n Enter the number of columns in Matrix 2 :");
		n=int.Parse(Console.ReadLine());
		b=new int[m,n];
		Console.WriteLine("\n Enter the elements of Matrix 2:");
		for(int i=0;i<b.GetLength(0);i++)
		{
			for(int j=0;j<b.GetLength(1);j++)
			{
				b[i,j]=int.Parse(Console.ReadLine());
			}
		}
	}
	
	public void PrintMatrix()
	{
		Console.WriteLine("\n Matrix 1:");
		for(int i=0;i<a.GetLength(0);i++)
		{
			for(int j=0;j<a.GetLength(1);j++)
			{
				Console.Write("\t"+a[i,j]);
			}
			Console.WriteLine();
		}
		Console.WriteLine("\n Matrix 2:");
		for(int i=0;i<b.GetLength(0);i++)
		{
			for(int j=0;j<b.GetLength(1);j++)
			{
				Console.Write("\t"+b[i,j]);
			}
			Console.WriteLine();
		}
		Console.WriteLine("\n Resultant Matrix after multiplying Matrix 1 & Matrix 2:");
		for(int i=0;i<c.GetLength(0);i++)
		{
			for(int j=0;j<c.GetLength(1);j++)
			{
				Console.Write("\t"+c[i,j]);
			}
			Console.WriteLine();
		}
		
	}
	public void MultiplyMatrix()
	{
		if(a.GetLength(1)==b.GetLength(0))
		{
			c=new int[a.GetLength(0),b.GetLength(1)];
			for(int i=0;i<c.GetLength(0);i++)
			{
				for(int j=0;j<c.GetLength(1);j++)
				{
					c[i,j]=0;
					for(int k=0;k<a.GetLength(1);k++) // OR k<b.GetLength(0)
					c[i,j]=c[i,j]+a[i,k]*b[k,j];
				}
			}
		}
		else
		{
			Console.WriteLine("\n Number of columns in Matrix1 is not equal to Number of rows in Matrix2.");
			Console.WriteLine("\n Therefore Multiplication of Matrix1 with Matrix2 is not possible");
			Environment.Exit(-1);
		}
	}
}
class Matrices
{
	public static void Main()
	{
		MatrixMultiplication MM=new MatrixMultiplication();
		MM.ReadMatrix();
		MM.MultiplyMatrix();
		MM.PrintMatrix();
	}
}
using System;
class MatrixMultiplication
{

int[,] a;
int[,] b;
int[,] c;
public void ReadMatrix()
{
Console.WriteLine("\n Size of Matrix 1:");
Console.Write("\n Enter the number of rows in Matrix 1 :");
int m=int.Parse(Console.ReadLine());
Console.Write("\n Enter the number of columns in Matrix 1 :");
int n=int.Parse(Console.ReadLine());
a=new int[m,n];
Console.WriteLine("\n Enter the elements of Matrix 1:");
for(int i=0;i<a.GetLength(0);i++)
{
for(int j=0;j<a.GetLength(1);j++)
{
a[i,j]=int.Parse(Console.ReadLine());
}
}

Console.WriteLine("\n Size of Matrix 2 :");
Console.Write("\n Enter the number of rows in Matrix 2 :");
m=int.Parse(Console.ReadLine());
Console.Write("\n Enter the number of columns in Matrix 2 :");
n=int.Parse(Console.ReadLine());
b=new int[m,n];
Console.WriteLine("\n Enter the elements of Matrix 2:");
for(int i=0;i<b.GetLength(0);i++)
{
for(int j=0;j<b.GetLength(1);j++)
{
b[i,j]=int.Parse(Console.ReadLine());
}
}
}
public void PrintMatrix()
{
Console.WriteLine("\n Matrix 1:");
for(int i=0;i<a.GetLength(0);i++)
{
for(int j=0;j<a.GetLength(1);j++)
{
Console.Write("\t"+a[i,j]);
}
Console.WriteLine();
}
Console.WriteLine("\n Matrix 2:");
for(int i=0;i<b.GetLength(0);i++)
{
for(int j=0;j<b.GetLength(1);j++)
{
Console.Write("\t"+b[i,j]);
}
Console.WriteLine();
}
Console.WriteLine("\n Resultant Matrix after multiplying Matrix 1 & Matrix 2:");
for(int i=0;i<c.GetLength(0);i++)
{
for(int j=0;j<c.GetLength(1);j++)
{
Console.Write("\t"+c[i,j]);
}
Console.WriteLine();
}

}
public void MultiplyMatrix()
{
if(a.GetLength(1)==b.GetLength(0))
{
c=new int[a.GetLength(0),b.GetLength(1)];
for(int i=0;i<c.GetLength(0);i++)
{
for(int j=0;j<c.GetLength(1);j++)
{
c[i,j]=0;
for(int k=0;k<a.GetLength(1);k++) // OR k<b.GetLength(0)
c[i,j]=c[i,j]+a[i,k]*b[k,j];
}
}
}
else
{
Console.WriteLine("\n Number of columns in Matrix1 is not equal to Number of rows in Matrix2.");
Console.WriteLine("\n Therefore Multiplication of Matrix1 with Matrix2 is not possible");
Environment.Exit(-1);
}
}
} class Matrices
{ public static void Main()
{
MatrixMultiplication MM=new MatrixMultiplication();
MM.ReadMatrix();
MM.MultiplyMatrix();
MM.PrintMatrix();
}
}

VB.NET Edit

Public Module MatrixMultiplication

    ' Generic stand-alone method
    ' Uses LINQ.
    Public Function Multiply(matrix1 As IEnumerable(Of IEnumerable(Of Double)), _
                             matrix2 As IEnumerable(Of IEnumerable(Of Double))
                    As IEnumerable(Of IEnumerable(Of Double))
        If matrix1.Count = matrix2(0).Count Then
            Dim ret(matrix1(0).Count, matrix2.Count) As Double
            For i As Integer = 0 To ret.Count - 1
                For j As Integer = 0 To ret(0).Count - 1
                    ret(i, j) = 0
                    For k As Integer = 0 To matrix1(0).Count - 1
                        ret(i, j) = ret(i, j) + a(i, k) + b(k, j)
                    Next
                Next
            Next
        Else
            Throw New RankException("Number of columns in matrix1 is not equal to the number of rows in matrix2.")
        End If
    End Function

    ' This method can be run in a console app.
    Public Sub Main()
        Dim m_1 As Integer, n_1 As Integer, m_2 As Integer, n_2 As Integer
        Try
            Console.WriteLine("Size of matrix 1:")
            Console.Write("Number of rows: ")
            m_1 = Integer.Parse(Console.ReadLine())
            Console.Write("Number of columns: ")
            n_1 = Integer.Parse(Console.ReadLine())
            Dim matrix1(m_1, n_1) As Double

            Console.WriteLine("Enter the elements of matrix 1:")
            For i As Integer = 0 To m_1 - 1
                For j As Integer = 0 To n_1 - 1
                    Console.Write("Element at " & i & ", " & j)
                    matrix1(i, j) = Integer.Parse(Console.ReadLine())
                Next
            Next
            
            Console.WriteLine("Size of matrix 2:")
            Console.Write("Number of rows: ")
            m_2 = Integer.Parse(Console.ReadLine())
            Console.Write("Number of columns: ")
            n_2 = Integer.Parse(Console.ReadLine())

            Console.WriteLine("Enter the elements of matrix 2:")
            For i As Integer = 0 To m_2 - 1
                For j As Integer = 0 To n_2 - 1
                    Console.Write("Element at " & i & ", " & j)
                    matrix2(i, j) = Integer.Parse(Console.ReadLine())
                Next
            Next

            Dim product(,) As Double = Multiply(matrix1, matrix2)

            Console.WriteLine("The product of the two matrices is:")
            For i As Integer = 0 To product.Count - 1
                For j As Integer = 0 To product(0).Count - 1
                    Console.Write(product(i, j) & " ")
                Next
                Console.WriteLine()
            Next
        Catch ex As Exception
            Console.WriteLine("An error occured. The message associated with the error was:")
            Console.WriteLine(ex.Message)
        End Try

        Console.WriteLine()
        Console.WriteLine("Press any key to exit.")
        Console.ReadKey()
    End Sub
End Module

Python Edit

This article is missing a code example in the Python language. 

def multmat(a,b):
	m=len(a)
	n=len(a[0])
	k=len(b)
	res=[]
	if len(b) != 1:
		p=len(b[0])-1
	else:
		p=0
	if len(b)==0:
		print('bad size')
	elif n!=k:
		print('bad size')
	else:
		#print('good size')
		n=k
		#print('k=',k,'n=',n)
		for q in range(m):
			res.append([0])
			#print('res=',res)
		for q in range(m):
			for w in range(p):
				res[q].append(0)
		for i in range(m):


			for j in range(p+1):


				for r in range(n):
					#print('ijr:res',i,j,r,res)

					res[i][j] =a[i][r]*b[r][j]+res[i][j]
	return res


+----------------+
| AN ALTERNATIVE |
+----------------+

...
s = 0
for i in range(self.rows):
    for c in range(other.cols):
        s = 0
        for j in range(other.cols):
            s += self.mat[i][j] * other.mat[j][c]
                    
    sum.mat[i][c] = s
...

Ruby Edit

This article is missing a code example in the Ruby language. 

require 'matrix'

def read_matrix
  puts "Enter number of column for matrix: "
  _cols = gets.chomp!.to_i
  
  puts "Enter number of rows for matrix: "
  _rows = gets.chomp!.to_i
  
  raise "Invalid sizes" unless _rows.is_a? Fixnum && _cols.is_a? Fixnum
  
  puts "Enter the elements of the matrix (one per line)"
  a = Matrix.build _rows, _cols, do |m|
    m = gets.chomp!.to_i
  end
  
  raise "An element is not a number" if a.any? { |eL| !eL.is_a? Fixnum }
  
  a
end

puts "reading first matrix..."
mat1 = read_matrix
puts "reading second matrix..."
mat2 = read_matrix

begin
  puts mat1 * mat2
rescue ExceptionForMatrix::ErrDimensionMismatch => e
  puts e.message
end



  # or if we don't care about user input
  # following the C++ example ..to show you how concise Ruby is
  
  Matrix.build(4) { rand } * Matrix.build(4) { rand }


This article is missing a code example in the Java language.

C++ Edit

int main(void) 
{
    const int N=4;
    int A[N][N], int B[N][N], int C[N][N];
    int i, j, k;
    int sum;
	for (i = 0; i < N; i++) 
	{
		for (j = 0; j < N; j++)
		{
			sum = 0;
			for (k = 0; k < N; k++)
			{
				sum += A[i][k] * B[k][j];
                        }
                         C[i][j] = sum;
                 }
        }
	return 0;
}
Retrieved from "http://code.wikia.com/wiki/Matrix_multiplication?oldid=8142"
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