Matrix multiplication
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Contents |
[edit] Introduction
[edit] Prerequisites
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
[edit] Matrix Multiplication Basics
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 
}. The resulting matrix will be of dimension n×n.
[edit] Example 4x2 Multiplied by 2x4
Take a matrix 4x2 call it 
; another of dimension 2x4 and call it 
. Give them some arbitrary values
and lets do some multiplication.
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:
As this implies, multiplication is non-commutative in general for matrices, i.e.
since in this case
if we reversed the order the resulting matrix 
would be
4×4 instead of 2×2.
[edit] More General Approach
Now lets visualize A and B as m×n and n×p matrices respectively.
We are going to be adding and multiplying like before, but generally.
[edit] Psedocode & General Algorithm
So now that we have a general idea of what a matrix is, and how to multiply them in general, we can derive some psedocode around it. We could break down the steps as follows.
- Check the sizes of two matrices (m×n) and (t×u): if n = t then we can multiply them otherwise no (in that order
)
- If they can be multiplied, then create a new square matrix of size n (or t, since n = t)
- For each row in A and each column in multiply and sum the elements and the place the results in the rows and columns of the result matrix
Here is some psedocode 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
[edit] Implementations
[edit] C#
// 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(); } }
